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| Boilerpipe Text | From Wikipedia, the free encyclopedia
"Hyperbolic curve" redirects here. For the geometric curve, see
Hyperbola
.
In
mathematics
,
hyperbolic functions
are analogues of the ordinary
trigonometric functions
, but defined using the
hyperbola
rather than the
circle
. Just as the points
(cos
t
, sin
t
)
form a
circle with a unit radius
, the points
(cosh
t
, sinh
t
)
form the right half of the
unit hyperbola
. Also, similarly to how the derivatives of
sin(
t
)
and
cos(
t
)
are
cos(
t
)
and
–sin(
t
)
respectively, the derivatives of
sinh(
t
)
and
cosh(
t
)
are
cosh(
t
)
and
sinh(
t
)
respectively.
Hyperbolic functions are used to express the
angle of parallelism
in
hyperbolic geometry
. They are used to express
Lorentz boosts
as
hyperbolic rotations
in
special relativity
. They also occur in the solutions of many linear
differential equations
(such as the equation defining a
catenary
),
cubic equations
, and
Laplace's equation
in
Cartesian coordinates
.
Laplace's equations
are important in many areas of
physics
, including
electromagnetic theory
,
heat transfer
, and
fluid dynamics
.
The basic hyperbolic functions are:
[
1
]
hyperbolic sine
"
sinh
" (
),
[
2
]
hyperbolic cosine
"
cosh
" (
),
[
3
]
from which are derived:
[
4
]
hyperbolic tangent
"
tanh
" (
),
[
5
]
hyperbolic cotangent
"
coth
" (
),
[
6
]
[
7
]
hyperbolic secant
"
sech
" (
),
[
8
]
hyperbolic cosecant
"
csch
" or "
cosech
" (
[
3
]
)
corresponding to the derived trigonometric functions.
The
inverse hyperbolic functions
are:
inverse hyperbolic sine
"
arsinh
" (also denoted "
sinh
−1
", "
asinh
" or sometimes "
arcsinh
")
[
9
]
[
10
]
[
11
]
inverse hyperbolic cosine
"
arcosh
" (also denoted "
cosh
−1
", "
acosh
" or sometimes "
arccosh
")
inverse hyperbolic tangent
"
artanh
" (also denoted "
tanh
−1
", "
atanh
" or sometimes "
arctanh
")
inverse hyperbolic cotangent
"
arcoth
" (also denoted "
coth
−1
", "
acoth
" or sometimes "
arccoth
")
inverse hyperbolic secant
"
arsech
" (also denoted "
sech
−1
", "
asech
" or sometimes "
arcsech
")
inverse hyperbolic cosecant
"
arcsch
" (also denoted "
arcosech
", "
csch
−1
", "
cosech
−1
","
acsch
", "
acosech
", or sometimes "
arccsch
" or "
arccosech
")
A
ray
through the
unit hyperbola
x
2
−
y
2
= 1
at the point
(cosh
a
, sinh
a
)
, where
a
is twice the area between the ray, the hyperbola, and the
x
-axis. For points on the hyperbola below the
x
-axis, the area is considered negative (see
animated version
with comparison with the trigonometric (circular) functions).
The hyperbolic functions take an
argument
called a
hyperbolic angle
. The magnitude of a hyperbolic angle is the
area
of its
hyperbolic sector
to
xy
= 1
. The hyperbolic functions may be defined in terms of the
legs of a right triangle
covering this sector.
In
complex analysis
, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are
entire functions
. As a result, the other hyperbolic functions are
meromorphic
in the whole complex plane.
By
Lindemann–Weierstrass theorem
, the hyperbolic functions have a
transcendental value
for every non-zero
algebraic value
of the argument.
[
12
]
The first known calculation of a hyperbolic trigonometry problem is attributed to
Gerardus Mercator
when issuing the
Mercator map projection
circa 1566. It requires tabulating solutions to a
transcendental equation
involving hyperbolic functions.
[
13
]
The first to suggest a similarity between the sector of the circle and that of the hyperbola was
Isaac Newton
in his 1687
Principia Mathematica
.
[
14
]
Roger Cotes
suggested to modify the trigonometric functions using the
imaginary unit
to obtain an oblate
spheroid
from a prolate one.
[
14
]
Hyperbolic functions were formally introduced in 1757 by
Vincenzo Riccati
.
[
14
]
[
13
]
[
15
]
Riccati used
Sc.
and
Cc.
(
sinus/cosinus circulare
) to refer to circular functions and
Sh.
and
Ch.
(
sinus/cosinus hyperbolico
) to refer to hyperbolic functions.
[
14
]
As early as 1759,
Daviet de Foncenex
showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended
de Moivre's formula
to hyperbolic functions.
[
15
]
[
14
]
During the 1760s,
Johann Heinrich Lambert
systematized the use functions and provided exponential expressions in various publications.
[
14
]
[
15
]
Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.
[
15
]
[
16
]
Right triangles with legs proportional to sinh and cosh
With
hyperbolic angle
u
, the hyperbolic functions sinh and cosh can be defined with the
exponential function
e
u
.
[
1
]
[
4
]
In the figure
.
Exponential definitions
[
edit
]
sinh
x
is half the
difference
of
e
x
and
e
−
x
cosh
x
is the
average
of
e
x
and
e
−
x
sinh
,
cosh
and
tanh
csch
,
sech
and
coth
Differential equation definitions
[
edit
]
The hyperbolic functions may be defined as solutions of
differential equations
: The hyperbolic sine and cosine are the solution
(
s
,
c
)
of the system
with the initial conditions
The initial conditions make the solution unique; without them any pair of functions
would be a solution.
sinh(
x
)
and
cosh(
x
)
are also the unique solution of the equation
f
″(
x
) =
f
(
x
)
,
such that
f
(0) = 1
,
f
′(0) = 0
for the hyperbolic cosine, and
f
(0) = 0
,
f
′(0) = 1
for the hyperbolic sine.
Complex trigonometric definitions
[
edit
]
Hyperbolic functions may also be deduced from
trigonometric functions
with
complex
arguments:
where
i
is the
imaginary unit
with
i
2
= −1
.
The above definitions are related to the exponential definitions via
Euler's formula
(See
§ Hyperbolic functions for complex numbers
below).
Characterizing properties
[
edit
]
It can be shown that the
area under the curve
of the hyperbolic cosine (over a finite interval) is always equal to the
arc length
corresponding to that interval:
[
17
]
The hyperbolic tangent is the (unique) solution to the
differential equation
f
′ = 1 −
f
2
, with
f
(0) = 0
.
[
18
]
[
19
]
The hyperbolic functions satisfy many identities, all of them similar in form to the
trigonometric identities
. In fact,
Osborn's rule
[
20
]
(named after
George Osborn
) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for
,
,
or
and
into a hyperbolic identity, by:
expanding it completely in terms of integral powers of sines and cosines,
changing sine to sinh and cosine to cosh, and
switching the sign of every term containing a product of two sinhs.
Odd
and
even
functions:
Reciprocals:
Analogous to
Euler's formula
:
Analogous to the
Pythagorean trigonometric identity
:
Sums and differences of arguments
[
edit
]
particularly
Addition and subtraction formulas
[
edit
]
Half argument formulas
[
edit
]
where
sgn
is the
sign function
.
If
x
≠ 0
then
Tangent half argument formulas
[
edit
]
When
,
The following inequality is useful in statistics:
[
21
]
It can be proved by comparing the Taylor series of the two functions term by term.
Inverse functions as logarithms
[
edit
]
Each of the functions
sinh
and
cosh
is equal to its
second derivative
, that is:
All functions with this property are
linear combinations
of
sinh
and
cosh
, in particular the
exponential functions
and
.
[
22
]
The following integrals can be proved using
hyperbolic substitution
:
where
C
is the
constant of integration
.
Taylor series expressions
[
edit
]
It is possible to express explicitly the
Taylor series
at zero (or the
Laurent series
, if the function is not defined at zero) of the above functions.
This series is
convergent
for every
complex
value of
x
. Since the function
sinh
x
is
odd
, only odd exponents for
x
occur in its Taylor series.
This series is
convergent
for every
complex
value of
x
. Since the function
cosh
x
is
even
, only even exponents for
x
occur in its Taylor series.
The sum of the sinh and cosh series is the
infinite series
expression of the
exponential function
.
The following series are followed by a description of a subset of their
domain of convergence
, where the series is convergent and its sum equals the function.
where:
Infinite products and continued fractions
[
edit
]
The following expansions are valid in the whole complex plane:
Comparison with circular functions
[
edit
]
Circle and hyperbola tangent at
(1, 1)
display geometry of circular functions in terms of
circular sector
area
u
and hyperbolic functions depending on
hyperbolic sector
area
u
.
The hyperbolic functions represent an expansion of
trigonometry
beyond the
circular functions
. Both types depend on an
argument
, either
circular angle
or
hyperbolic angle
.
Since the
area of a circular sector
with radius
r
and angle
u
(in radians) is
r
2
u
/2
, it will be equal to
u
when
r
=
√
2
. In the diagram, such a circle is tangent to the hyperbola
xy
= 1
at
(1, 1)
. The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a
hyperbolic sector
with area corresponding to hyperbolic angle magnitude.
The legs of the two
right triangles
with the
hypotenuse
on the ray defining the angles are of length
√
2
times the circular and hyperbolic functions.
The hyperbolic angle is an
invariant measure
with respect to the
squeeze mapping
, just as the circular angle is invariant under rotation.
[
23
]
The
Gudermannian function
gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.
The graph of the function
is the
catenary
, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
Relationship to the exponential function
[
edit
]
The decomposition of the exponential function in its
even and odd parts
gives the identities
and
Combined with
Euler's formula
this gives
for the
general complex exponential function
.
Additionally,
Hyperbolic functions for complex numbers
[
edit
]
Hyperbolic functions in the complex plane
Since the
exponential function
can be defined for any
complex
argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions
sinh
z
and
cosh
z
are then
holomorphic
.
Relationships to ordinary trigonometric functions are given by
Euler's formula
for complex numbers:
so:
Thus, hyperbolic functions are
periodic
with respect to the imaginary component, with period
(
for hyperbolic tangent and cotangent).
e (mathematical constant)
Equal incircles theorem
, based on sinh
Hyperbolastic functions
Hyperbolic growth
Inverse hyperbolic functions
List of integrals of hyperbolic functions
Poinsot's spirals
Sigmoid function
Trigonometric functions
^
a
b
c
d
Weisstein, Eric W.
"Hyperbolic Functions"
.
mathworld.wolfram.com
. Retrieved
2020-08-29
.
^
(1999)
Collins Concise Dictionary
, 4th edition, HarperCollins, Glasgow,
ISBN
0 00 472257 4
, p. 1386
^
a
b
Collins Concise Dictionary
, p. 328
^
a
b
"Hyperbolic Functions"
.
www.mathsisfun.com
. Retrieved
2020-08-29
.
^
Collins Concise Dictionary
, p. 1520
^
Collins Concise Dictionary
, p. 329
^
tanh
^
Collins Concise Dictionary
, p. 1340
^
Woodhouse, N. M. J.
(2003),
Special Relativity
, London: Springer, p. 71,
ISBN
978-1-85233-426-0
^
Abramowitz, Milton
;
Stegun, Irene A.
, eds. (1972),
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
, New York:
Dover Publications
,
ISBN
978-0-486-61272-0
^
Some examples of using
arcsinh
found in
Google Books
.
^
Niven, Ivan (1985).
Irrational Numbers
. Vol. 11. Mathematical Association of America.
ISBN
9780883850381
.
JSTOR
10.4169/j.ctt5hh8zn
.
^
a
b
George F. Becker; C. E. Van Orstrand (1909).
Hyperbolic Functions
. Universal Digital Library. The Smithsonian Institution.
^
a
b
c
d
e
f
McMahon, James (1896).
Hyperbolic Functions
. Osmania University, Digital Library Of India. John Wiley And Sons.
^
a
b
c
d
Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward.
Euler at 300: an appreciation.
Mathematical Association of America, 2007. Page 100.
^
Becker, Georg F.
Hyperbolic functions.
Read Books, 1931. Page xlviii.
^
N.P., Bali (2005).
Golden Integral Calculus
. Firewall Media. p. 472.
ISBN
81-7008-169-6
.
^
Steeb, Willi-Hans (2005).
Nonlinear Workbook, The: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic With C++, Java And Symbolicc++ Programs
(3rd ed.). World Scientific Publishing Company. p. 281.
ISBN
978-981-310-648-2
.
Extract of page 281 (using lambda=1)
^
Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010).
An Atlas of Functions: with Equator, the Atlas Function Calculator
(2nd, illustrated ed.). Springer Science & Business Media. p. 290.
ISBN
978-0-387-48807-3
.
Extract of page 290
^
Osborn, G. (July 1902).
"Mnemonic for hyperbolic formulae"
.
The Mathematical Gazette
.
2
(34): 189.
doi
:
10.2307/3602492
.
JSTOR
3602492
.
S2CID
125866575
.
^
Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627.
[1]
^
Olver, Frank W. J.
; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010),
"Hyperbolic functions"
,
NIST Handbook of Mathematical Functions
, Cambridge University Press,
ISBN
978-0-521-19225-5
,
MR
2723248
.
^
Haskell, Mellen W.
, "On the introduction of the notion of hyperbolic functions",
Bulletin of the American Mathematical Society
1
:6:155–9,
full text
"Hyperbolic functions"
,
Encyclopedia of Mathematics
,
EMS Press
, 2001 [1994]
Hyperbolic functions
on
PlanetMath
GonioLab
: Visualization of the unit circle, trigonometric and hyperbolic functions (
Java Web Start
)
Web-based calculator of hyperbolic functions |
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## Contents
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- [1 History](https://en.wikipedia.org/wiki/Hyperbolic_functions#History)
- [2 Notation](https://en.wikipedia.org/wiki/Hyperbolic_functions#Notation)
- [3 Definitions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Definitions)
Toggle Definitions subsection
- [3\.1 Exponential definitions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Exponential_definitions)
- [3\.2 Differential equation definitions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Differential_equation_definitions)
- [3\.3 Complex trigonometric definitions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Complex_trigonometric_definitions)
- [4 Characterizing properties](https://en.wikipedia.org/wiki/Hyperbolic_functions#Characterizing_properties)
Toggle Characterizing properties subsection
- [4\.1 Hyperbolic cosine](https://en.wikipedia.org/wiki/Hyperbolic_functions#Hyperbolic_cosine)
- [4\.2 Hyperbolic tangent](https://en.wikipedia.org/wiki/Hyperbolic_functions#Hyperbolic_tangent)
- [5 Useful relations](https://en.wikipedia.org/wiki/Hyperbolic_functions#Useful_relations)
Toggle Useful relations subsection
- [5\.1 Sums and differences of arguments](https://en.wikipedia.org/wiki/Hyperbolic_functions#Sums_and_differences_of_arguments)
- [5\.2 Addition and subtraction formulas](https://en.wikipedia.org/wiki/Hyperbolic_functions#Addition_and_subtraction_formulas)
- [5\.3 Product formulas](https://en.wikipedia.org/wiki/Hyperbolic_functions#Product_formulas)
- [5\.4 Half argument formulas](https://en.wikipedia.org/wiki/Hyperbolic_functions#Half_argument_formulas)
- [5\.5 Tangent half argument formulas](https://en.wikipedia.org/wiki/Hyperbolic_functions#Tangent_half_argument_formulas)
- [5\.6 Square formulas](https://en.wikipedia.org/wiki/Hyperbolic_functions#Square_formulas)
- [5\.7 Inequalities](https://en.wikipedia.org/wiki/Hyperbolic_functions#Inequalities)
- [6 Inverse functions as logarithms](https://en.wikipedia.org/wiki/Hyperbolic_functions#Inverse_functions_as_logarithms)
- [7 Derivatives](https://en.wikipedia.org/wiki/Hyperbolic_functions#Derivatives)
- [8 Second derivatives](https://en.wikipedia.org/wiki/Hyperbolic_functions#Second_derivatives)
- [9 Standard integrals](https://en.wikipedia.org/wiki/Hyperbolic_functions#Standard_integrals)
- [10 Taylor series expressions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Taylor_series_expressions)
- [11 Infinite products and continued fractions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Infinite_products_and_continued_fractions)
- [12 Comparison with circular functions](https://en.wikipedia.org/wiki/Hyperbolic_functions#Comparison_with_circular_functions)
- [13 Relationship to the exponential function](https://en.wikipedia.org/wiki/Hyperbolic_functions#Relationship_to_the_exponential_function)
- [14 Hyperbolic functions for complex numbers](https://en.wikipedia.org/wiki/Hyperbolic_functions#Hyperbolic_functions_for_complex_numbers)
- [15 See also](https://en.wikipedia.org/wiki/Hyperbolic_functions#See_also)
- [16 References](https://en.wikipedia.org/wiki/Hyperbolic_functions#References)
- [17 External links](https://en.wikipedia.org/wiki/Hyperbolic_functions#External_links)
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# Hyperbolic functions
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- [Čeština](https://cs.wikipedia.org/wiki/Hyperbolick%C3%A9_funkce "Hyperbolické funkce – Czech")
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- [Dansk](https://da.wikipedia.org/wiki/Hyperbolske_funktioner "Hyperbolske funktioner – Danish")
- [Deutsch](https://de.wikipedia.org/wiki/Hyperbelfunktion "Hyperbelfunktion – German")
- [Ελληνικά](https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%B5%CF%81%CE%B2%CE%BF%CE%BB%CE%B9%CE%BA%CE%AD%CF%82_%CF%83%CF%85%CE%BD%CE%B1%CF%81%CF%84%CE%AE%CF%83%CE%B5%CE%B9%CF%82 "Υπερβολικές συναρτήσεις – Greek")
- [Esperanto](https://eo.wikipedia.org/wiki/Hiperbola_funkcio "Hiperbola funkcio – Esperanto")
- [Español](https://es.wikipedia.org/wiki/Funci%C3%B3n_hiperb%C3%B3lica "Función hiperbólica – Spanish")
- [Euskara](https://eu.wikipedia.org/wiki/Funtzio_hiperboliko "Funtzio hiperboliko – Basque")
- [فارسی](https://fa.wikipedia.org/wiki/%D8%AA%D8%A7%D8%A8%D8%B9_%D9%87%D8%B0%D9%84%D9%88%D9%84%D9%88%DB%8C "تابع هذلولوی – Persian")
- [Suomi](https://fi.wikipedia.org/wiki/Hyperbolinen_funktio "Hyperbolinen funktio – Finnish")
- [Français](https://fr.wikipedia.org/wiki/Fonction_hyperbolique "Fonction hyperbolique – French")
- [Gaeilge](https://ga.wikipedia.org/wiki/Feidhmeanna_hipearb%C3%B3ileacha "Feidhmeanna hipearbóileacha – Irish")
- [Galego](https://gl.wikipedia.org/wiki/Funci%C3%B3n_hiperb%C3%B3lica "Función hiperbólica – Galician")
- [עברית](https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA_%D7%94%D7%99%D7%A4%D7%A8%D7%91%D7%95%D7%9C%D7%99%D7%95%D7%AA "פונקציות היפרבוליות – Hebrew")
- [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%A4%E0%A4%BF%E0%A4%AA%E0%A4%B0%E0%A4%B5%E0%A4%B2%E0%A4%AF%E0%A4%BF%E0%A4%95_%E0%A4%AB%E0%A4%B2%E0%A4%A8 "अतिपरवलयिक फलन – Hindi")
- [Hrvatski](https://hr.wikipedia.org/wiki/Hiperbolne_funkcije "Hiperbolne funkcije – Croatian")
- [Magyar](https://hu.wikipedia.org/wiki/Hiperbolikus_f%C3%BCggv%C3%A9nyek "Hiperbolikus függvények – Hungarian")
- [Հայերեն](https://hy.wikipedia.org/wiki/%D5%80%D5%AB%D5%BA%D5%A5%D6%80%D5%A2%D5%B8%D5%AC%D5%A1%D5%AF%D5%A1%D5%B6_%D6%86%D5%B8%D6%82%D5%B6%D5%AF%D6%81%D5%AB%D5%A1%D5%B6%D5%A5%D6%80 "Հիպերբոլական ֆունկցիաներ – Armenian")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Fungsi_hiperbolik "Fungsi hiperbolik – Indonesian")
- [Íslenska](https://is.wikipedia.org/wiki/Brei%C3%B0bogafall "Breiðbogafall – Icelandic")
- [Italiano](https://it.wikipedia.org/wiki/Funzioni_iperboliche "Funzioni iperboliche – Italian")
- [日本語](https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A%E9%96%A2%E6%95%B0 "双曲線関数 – Japanese")
- [ភាសាខ្មែរ](https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%93%E1%9E%BB%E1%9E%82%E1%9E%98%E1%9E%93%E1%9F%8D%E1%9E%A2%E1%9F%8A%E1%9E%B8%E1%9E%96%E1%9F%82%E1%9E%94%E1%9E%BC%E1%9E%9B%E1%9E%B8%E1%9E%80 "អនុគមន៍អ៊ីពែបូលីក – Khmer")
- [한국어](https://ko.wikipedia.org/wiki/%EC%8C%8D%EA%B3%A1%EC%84%A0_%ED%95%A8%EC%88%98 "쌍곡선 함수 – Korean")
- [Latina](https://la.wikipedia.org/wiki/Functiones_hyperbolicae "Functiones hyperbolicae – Latin")
- [Latviešu](https://lv.wikipedia.org/wiki/Hiperbolisk%C4%81s_funkcijas "Hiperboliskās funkcijas – Latvian")
- [Македонски](https://mk.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B8_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8 "Хиперболични функции – Macedonian")
- [Bahasa Melayu](https://ms.wikipedia.org/wiki/Fungsi_hiperbola "Fungsi hiperbola – Malay")
- [Nederlands](https://nl.wikipedia.org/wiki/Hyperbolische_functie "Hyperbolische functie – Dutch")
- [Norsk nynorsk](https://nn.wikipedia.org/wiki/Hyperbolsk_funksjon "Hyperbolsk funksjon – Norwegian Nynorsk")
- [Norsk bokmål](https://no.wikipedia.org/wiki/Hyperbolsk_funksjon "Hyperbolsk funksjon – Norwegian Bokmål")
- [Polski](https://pl.wikipedia.org/wiki/Funkcje_hiperboliczne "Funkcje hiperboliczne – Polish")
- [Português](https://pt.wikipedia.org/wiki/Fun%C3%A7%C3%A3o_hiperb%C3%B3lica "Função hiperbólica – Portuguese")
- [Română](https://ro.wikipedia.org/wiki/Func%C8%9Bie_hiperbolic%C4%83 "Funcție hiperbolică – Romanian")
- [Русский](https://ru.wikipedia.org/wiki/%D0%93%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D0%B8 "Гиперболические функции – Russian")
- [Srpskohrvatski / српскохрватски](https://sh.wikipedia.org/wiki/Hiperboli%C4%8Dne_funkcije "Hiperbolične funkcije – Serbo-Croatian")
- [සිංහල](https://si.wikipedia.org/wiki/%E0%B6%B6%E0%B7%84%E0%B7%94%E0%B7%80%E0%B6%BD%E0%B6%BA%E0%B7%92%E0%B6%9A_%E0%B7%81%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B7%92%E0%B6%AD "බහුවලයික ශ්රිත – Sinhala")
- [Simple English](https://simple.wikipedia.org/wiki/Hyperbolic_functions "Hyperbolic functions – Simple English")
- [Slovenčina](https://sk.wikipedia.org/wiki/Hyperbolick%C3%A1_funkcia "Hyperbolická funkcia – Slovak")
- [Slovenščina](https://sl.wikipedia.org/wiki/Hiperboli%C4%8Dna_funkcija "Hiperbolična funkcija – Slovenian")
- [Shqip](https://sq.wikipedia.org/wiki/Funksionet_hiperbolike "Funksionet hiperbolike – Albanian")
- [Српски / srpski](https://sr.wikipedia.org/wiki/%D0%A5%D0%B8%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D0%B8%D1%87%D0%BD%D0%B5_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D0%B8%D1%98%D0%B5 "Хиперболичне функције – Serbian")
- [Svenska](https://sv.wikipedia.org/wiki/Hyperbolisk_funktion "Hyperbolisk funktion – Swedish")
- [தமிழ்](https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%9A%E0%AF%8D_%E0%AE%9A%E0%AE%BE%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%81 "அதிபரவளையச் சார்பு – Tamil")
- [Tagalog](https://tl.wikipedia.org/wiki/Punsiyong_hiperboliko "Punsiyong hiperboliko – Tagalog")
- [Türkçe](https://tr.wikipedia.org/wiki/Hiperbolik_fonksiyon "Hiperbolik fonksiyon – Turkish")
- [Українська](https://uk.wikipedia.org/wiki/%D0%93%D1%96%D0%BF%D0%B5%D1%80%D0%B1%D0%BE%D0%BB%D1%96%D1%87%D0%BD%D1%96_%D1%84%D1%83%D0%BD%D0%BA%D1%86%D1%96%D1%97 "Гіперболічні функції – Ukrainian")
- [Oʻzbekcha / ўзбекча](https://uz.wikipedia.org/wiki/Giperbolik_funksiyalar "Giperbolik funksiyalar – Uzbek")
- [Tiếng Việt](https://vi.wikipedia.org/wiki/H%C3%A0m_hyperbol "Hàm hyperbol – Vietnamese")
- [吴语](https://wuu.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E5%87%BD%E6%95%B0 "双曲函数 – Wu")
- [閩南語 / Bân-lâm-gí](https://zh-min-nan.wikipedia.org/wiki/Siang-khiok_h%C3%A2m-s%C3%B2%CD%98 "Siang-khiok hâm-sò͘ – Minnan")
- [粵語](https://zh-yue.wikipedia.org/wiki/%E9%9B%99%E6%9B%B2%E5%87%BD%E6%95%B8 "雙曲函數 – Cantonese")
- [中文](https://zh.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E5%87%BD%E6%95%B0 "双曲函数 – Chinese")
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From Wikipedia, the free encyclopedia
Hyperbolic analogues of trigonometric functions
"Hyperbolic curve" redirects here. For the geometric curve, see [Hyperbola](https://en.wikipedia.org/wiki/Hyperbola "Hyperbola").
[](https://en.wikipedia.org/wiki/File:Sinh_cosh_tanh.svg)
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), **hyperbolic functions** are analogues of the ordinary [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function"), but defined using the [hyperbola](https://en.wikipedia.org/wiki/Hyperbola "Hyperbola") rather than the [circle](https://en.wikipedia.org/wiki/Circle "Circle"). Just as the points (cos *t*, sin *t*) form a [circle with a unit radius](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"), the points (cosh *t*, sinh *t*) form the right half of the [unit hyperbola](https://en.wikipedia.org/wiki/Unit_hyperbola "Unit hyperbola"). Also, similarly to how the derivatives of sin(*t*) and cos(*t*) are cos(*t*) and –sin(*t*) respectively, the derivatives of sinh(*t*) and cosh(*t*) are cosh(*t*) and sinh(*t*) respectively.
Hyperbolic functions are used to express the [angle of parallelism](https://en.wikipedia.org/wiki/Angle_of_parallelism "Angle of parallelism") in [hyperbolic geometry](https://en.wikipedia.org/wiki/Hyperbolic_geometry "Hyperbolic geometry"). They are used to express [Lorentz boosts](https://en.wikipedia.org/wiki/Lorentz_boost "Lorentz boost") as [hyperbolic rotations](https://en.wikipedia.org/wiki/Hyperbolic_rotation "Hyperbolic rotation") in [special relativity](https://en.wikipedia.org/wiki/Special_relativity "Special relativity"). They also occur in the solutions of many linear [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") (such as the equation defining a [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary")), [cubic equations](https://en.wikipedia.org/wiki/Cubic_equation#Hyperbolic_solution_for_one_real_root "Cubic equation"), and [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") in [Cartesian coordinates](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"). [Laplace's equations](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") are important in many areas of [physics](https://en.wikipedia.org/wiki/Physics "Physics"), including [electromagnetic theory](https://en.wikipedia.org/wiki/Electromagnetic_theory "Electromagnetic theory"), [heat transfer](https://en.wikipedia.org/wiki/Heat_transfer "Heat transfer"), and [fluid dynamics](https://en.wikipedia.org/wiki/Fluid_dynamics "Fluid dynamics").
The basic hyperbolic functions are:[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)
- **hyperbolic sine** "sinh" ([/ˈsɪŋ, ˈsɪntʃ, ˈʃaɪn/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")),[\[2\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-2)
- **hyperbolic cosine** "cosh" ([/ˈkɒʃ, ˈkoʊʃ/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")),[\[3\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Collins_Concise_Dictionary_p._328-3)
from which are derived:[\[4\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:2-4)
- **hyperbolic tangent** "tanh" ([/ˈtæŋ, ˈtæntʃ, ˈθæn/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")),[\[5\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-5)
- **hyperbolic cotangent** "coth" ([/ˈkɒθ, ˈkoʊθ/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")),[\[6\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-6)[\[7\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-7)
- **hyperbolic secant** "sech" ([/ˈsɛtʃ, ˈʃɛk/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")),[\[8\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-8)
- **hyperbolic cosecant** "csch" or "cosech" ([/ˈkoʊsɛtʃ, ˈkoʊʃɛk/](https://en.wikipedia.org/wiki/Help:IPA/English "Help:IPA/English")[\[3\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Collins_Concise_Dictionary_p._328-3))
corresponding to the derived trigonometric functions.
The [inverse hyperbolic functions](https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions "Inverse hyperbolic functions") are:
- **inverse hyperbolic sine** "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[\[9\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-9)[\[10\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-10)[\[11\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-11)
- **inverse hyperbolic cosine** "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")
- **inverse hyperbolic tangent** "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh")
- **inverse hyperbolic cotangent** "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth")
- **inverse hyperbolic secant** "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech")
- **inverse hyperbolic cosecant** "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech")
[](https://en.wikipedia.org/wiki/File:Hyperbolic_functions-2.svg)
A [ray](https://en.wikipedia.org/wiki/Ray_\(geometry\) "Ray (geometry)") through the [unit hyperbola](https://en.wikipedia.org/wiki/Unit_hyperbola "Unit hyperbola") *x*2 − *y*2 = 1 at the point (cosh *a*, sinh *a*), where a is twice the area between the ray, the hyperbola, and the x\-axis. For points on the hyperbola below the x\-axis, the area is considered negative (see [animated version](https://en.wikipedia.org/wiki/File:HyperbolicAnimation.gif "File:HyperbolicAnimation.gif") with comparison with the trigonometric (circular) functions).
The hyperbolic functions take an [argument](https://en.wikipedia.org/wiki/Argument_of_a_function "Argument of a function") called a [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle"). The magnitude of a hyperbolic angle is the [area](https://en.wikipedia.org/wiki/Area "Area") of its [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") to *xy* = 1. The hyperbolic functions may be defined in terms of the [legs of a right triangle](https://en.wikipedia.org/wiki/Hyperbolic_sector#Hyperbolic_triangle "Hyperbolic sector") covering this sector.
In [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are [entire functions](https://en.wikipedia.org/wiki/Entire_function "Entire function"). As a result, the other hyperbolic functions are [meromorphic](https://en.wikipedia.org/wiki/Meromorphic_function "Meromorphic function") in the whole complex plane.
By [Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem "Lindemann–Weierstrass theorem"), the hyperbolic functions have a [transcendental value](https://en.wikipedia.org/wiki/Transcendental_number "Transcendental number") for every non-zero [algebraic value](https://en.wikipedia.org/wiki/Algebraic_number "Algebraic number") of the argument.[\[12\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-12)
## History
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=1 "Edit section: History")\]
The first known calculation of a hyperbolic trigonometry problem is attributed to [Gerardus Mercator](https://en.wikipedia.org/wiki/Gerardus_Mercator "Gerardus Mercator") when issuing the [Mercator map projection](https://en.wikipedia.org/wiki/Mercator_projection "Mercator projection") circa 1566. It requires tabulating solutions to a [transcendental equation](https://en.wikipedia.org/wiki/Transcendental_equation "Transcendental equation") involving hyperbolic functions.[\[13\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:3-13)
The first to suggest a similarity between the sector of the circle and that of the hyperbola was [Isaac Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") in his 1687 [*Principia Mathematica*](https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica "Philosophiæ Naturalis Principia Mathematica").[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
[Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "Roger Cotes") suggested to modify the trigonometric functions using the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit") i \= − 1 {\\displaystyle i={\\sqrt {-1}}}  to obtain an oblate [spheroid](https://en.wikipedia.org/wiki/Spheroid "Spheroid") from a prolate one.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
Hyperbolic functions were formally introduced in 1757 by [Vincenzo Riccati](https://en.wikipedia.org/wiki/Vincenzo_Riccati "Vincenzo Riccati").[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)[\[13\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:3-13)[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15) Riccati used *Sc.* and *Cc.* (*sinus/cosinus circulare*) to refer to circular functions and *Sh.* and *Ch.* (*sinus/cosinus hyperbolico*) to refer to hyperbolic functions.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14) As early as 1759, [Daviet de Foncenex](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Daviet_de_Foncenex "François Daviet de Foncenex") showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended [de Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula") to hyperbolic functions.[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15)[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
During the 1760s, [Johann Heinrich Lambert](https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert "Johann Heinrich Lambert") systematized the use functions and provided exponential expressions in various publications.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15) Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15)[\[16\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-16)
## Notation
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=2 "Edit section: Notation")\]
Main article: [Trigonometric functions § Notation](https://en.wikipedia.org/wiki/Trigonometric_functions#Notation "Trigonometric functions")
## Definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=3 "Edit section: Definitions")\]
[](https://en.wikipedia.org/wiki/File:Cartesian_hyperbolic_rhombus.svg)
Right triangles with legs proportional to sinh and cosh
With [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle") *u*, the hyperbolic functions sinh and cosh can be defined with the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") eu.[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)[\[4\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:2-4) In the figure A \= ( e − u , e u ) , B \= ( e u , e − u ) , O A \+ O B \= O C {\\displaystyle A=(e^{-u},e^{u}),\\ B=(e^{u},\\ e^{-u}),\\ OA+OB=OC}  .
### Exponential definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=4 "Edit section: Exponential definitions")\]
[](https://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_sinh.svg)
sinh *x* is half the [difference](https://en.wikipedia.org/wiki/Subtraction "Subtraction") of *ex* and *e*−*x*
[](https://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_cosh.svg)
cosh *x* is the [average](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean") of *ex* and *e*−*x*
- Hyperbolic sine: the [odd part](https://en.wikipedia.org/wiki/Odd_part_of_a_function "Odd part of a function") of the exponential function, that is,
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{\\displaystyle \\sinh x={\\frac {e^{x}-e^{-x}}{2}}={\\frac {e^{2x}-1}{2e^{x}}}.}
- Hyperbolic cosine: the [even part](https://en.wikipedia.org/wiki/Even_part_of_a_function "Even part of a function") of the exponential function, that is,
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{\\displaystyle \\cosh x={\\frac {e^{x}+e^{-x}}{2}}={\\frac {e^{2x}+1}{2e^{x}}}.}
[](https://en.wikipedia.org/wiki/File:Sinh_cosh_tanh.svg)
sinh, cosh and tanh
[](https://en.wikipedia.org/wiki/File:Csch_sech_coth.svg)
csch, sech and coth
- Hyperbolic tangent:
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{\\displaystyle \\tanh x={\\frac {\\sinh x}{\\cosh x}}={\\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}={\\frac {e^{2x}-1}{e^{2x}+1}}.}
- Hyperbolic cotangent: for *x* ≠ 0,
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{\\displaystyle \\coth x={\\frac {\\cosh x}{\\sinh x}}={\\frac {e^{x}+e^{-x}}{e^{x}-e^{-x}}}={\\frac {e^{2x}+1}{e^{2x}-1}}.}
- Hyperbolic secant:
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{\\displaystyle \\operatorname {sech} x={\\frac {1}{\\cosh x}}={\\frac {2}{e^{x}+e^{-x}}}={\\frac {2e^{x}}{e^{2x}+1}}.}
- Hyperbolic cosecant: for *x* ≠ 0,
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{\\displaystyle \\operatorname {csch} x={\\frac {1}{\\sinh x}}={\\frac {2}{e^{x}-e^{-x}}}={\\frac {2e^{x}}{e^{2x}-1}}.}
### Differential equation definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=5 "Edit section: Differential equation definitions")\]
The hyperbolic functions may be defined as solutions of [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"): The hyperbolic sine and cosine are the solution (*s*, *c*) of the system c ′ ( x ) \= s ( x ) , s ′ ( x ) \= c ( x ) , {\\displaystyle {\\begin{aligned}c'(x)&=s(x),\\\\s'(x)&=c(x),\\\\\\end{aligned}}}  with the initial conditions s ( 0 ) \= 0 , c ( 0 ) \= 1\. {\\displaystyle s(0)=0,c(0)=1.}  The initial conditions make the solution unique; without them any pair of functions ( a e x \+ b e − x , a e x − b e − x ) {\\displaystyle (ae^{x}+be^{-x},ae^{x}-be^{-x})}  would be a solution.
sinh(*x*) and cosh(*x*) are also the unique solution of the equation *f* ″(*x*) = *f* (*x*), such that *f* (0) = 1, *f* ′(0) = 0 for the hyperbolic cosine, and *f* (0) = 0, *f* ′(0) = 1 for the hyperbolic sine.
### Complex trigonometric definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=6 "Edit section: Complex trigonometric definitions")\]
Hyperbolic functions may also be deduced from [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function") with [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") arguments:
- Hyperbolic sine:[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)
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{\\displaystyle \\sinh x=-i\\sin(ix).}
- Hyperbolic cosine:[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)
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{\\displaystyle \\cosh x=\\cos(ix).}
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{\\displaystyle \\tanh x=-i\\tan(ix).}
- Hyperbolic cotangent:
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{\\displaystyle \\coth x=i\\cot(ix).}
- Hyperbolic secant:
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{\\displaystyle \\operatorname {sech} x=\\sec(ix).}
- Hyperbolic cosecant:
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{\\displaystyle \\operatorname {csch} x=i\\csc(ix).}
where i is the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit") with *i*2 = −1.
The above definitions are related to the exponential definitions via [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") (See [§ Hyperbolic functions for complex numbers](https://en.wikipedia.org/wiki/Hyperbolic_functions#Hyperbolic_functions_for_complex_numbers) below).
## Characterizing properties
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=7 "Edit section: Characterizing properties")\]
### Hyperbolic cosine
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=8 "Edit section: Hyperbolic cosine")\]
It can be shown that the [area under the curve](https://en.wikipedia.org/wiki/Area_under_the_curve "Area under the curve") of the hyperbolic cosine (over a finite interval) is always equal to the [arc length](https://en.wikipedia.org/wiki/Arc_length "Arc length") corresponding to that interval:[\[17\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-17) area \= ∫ a b cosh x d x \= ∫ a b 1 \+ ( d d x cosh x ) 2 d x \= arc length. {\\displaystyle {\\text{area}}=\\int \_{a}^{b}\\cosh x\\,dx=\\int \_{a}^{b}{\\sqrt {1+\\left({\\frac {d}{dx}}\\cosh x\\right)^{2}}}\\,dx={\\text{arc length.}}} 
### Hyperbolic tangent
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=9 "Edit section: Hyperbolic tangent")\]
The hyperbolic tangent is the (unique) solution to the [differential equation](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") *f* ′ = 1 − *f* 2, with *f* (0) = 0.[\[18\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-18)[\[19\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-19)
## Useful relations
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=10 "Edit section: Useful relations")\]
The hyperbolic functions satisfy many identities, all of them similar in form to the [trigonometric identities](https://en.wikipedia.org/wiki/Trigonometric_identity "Trigonometric identity"). In fact, **Osborn's rule**[\[20\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Osborn,_1902-20) (named after [George Osborn](https://en.wikipedia.org/wiki/George_Osborn_\(mathematician\) "George Osborn (mathematician)")) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for θ {\\displaystyle \\theta } , 2 θ {\\displaystyle 2\\theta } , 3 θ {\\displaystyle 3\\theta }  or θ {\\displaystyle \\theta }  and φ {\\displaystyle \\varphi }  into a hyperbolic identity, by:
1. expanding it completely in terms of integral powers of sines and cosines,
2. changing sine to sinh and cosine to cosh, and
3. switching the sign of every term containing a product of two sinhs.
[Odd](https://en.wikipedia.org/wiki/Odd_function "Odd function") and [even](https://en.wikipedia.org/wiki/Even_function "Even function") functions: sinh ( − x ) \= − sinh x cosh ( − x ) \= cosh x tanh ( − x ) \= − tanh x coth ( − x ) \= − coth x sech ( − x ) \= sech x csch ( − x ) \= − csch x {\\displaystyle {\\begin{aligned}\\sinh(-x)&=-\\sinh x\\\\\\cosh(-x)&=\\cosh x\\\\\\tanh(-x)&=-\\tanh x\\\\\\coth(-x)&=-\\coth x\\\\\\operatorname {sech} (-x)&=\\operatorname {sech} x\\\\\\operatorname {csch} (-x)&=-\\operatorname {csch} x\\end{aligned}}} 
Reciprocals:
arsech x \= arcosh ( 1 x ) arcsch x \= arsinh ( 1 x ) arcoth x \= artanh ( 1 x ) {\\displaystyle {\\begin{aligned}\\operatorname {arsech} x&=\\operatorname {arcosh} \\left({\\frac {1}{x}}\\right)\\\\\\operatorname {arcsch} x&=\\operatorname {arsinh} \\left({\\frac {1}{x}}\\right)\\\\\\operatorname {arcoth} x&=\\operatorname {artanh} \\left({\\frac {1}{x}}\\right)\\end{aligned}}} 
Analogous to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"):
cosh x \+ sinh x \= e x cosh x − sinh x \= e − x {\\displaystyle {\\begin{aligned}\\cosh x+\\sinh x&=e^{x}\\\\\\cosh x-\\sinh x&=e^{-x}\\end{aligned}}} 
Analogous to the [Pythagorean trigonometric identity](https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity "Pythagorean trigonometric identity"):
cosh 2 x − sinh 2 x \= 1 1 − tanh 2 x \= sech 2 x coth 2 x − 1 \= csch 2 x {\\displaystyle {\\begin{aligned}\\cosh ^{2}x-\\sinh ^{2}x&=1\\\\1-\\tanh ^{2}x&=\\operatorname {sech} ^{2}x\\\\\\coth ^{2}x-1&=\\operatorname {csch} ^{2}x\\end{aligned}}} 
### Sums and differences of arguments
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=11 "Edit section: Sums and differences of arguments")\]
sinh ( x \+ y ) \= sinh x cosh y \+ cosh x sinh y cosh ( x \+ y ) \= cosh x cosh y \+ sinh x sinh y tanh ( x \+ y ) \= tanh x \+ tanh y 1 \+ tanh x tanh y sinh ( x − y ) \= sinh x cosh y − cosh x sinh y cosh ( x − y ) \= cosh x cosh y − sinh x sinh y tanh ( x − y ) \= tanh x − tanh y 1 − tanh x tanh y {\\displaystyle {\\begin{aligned}\\sinh(x+y)&=\\sinh x\\cosh y+\\cosh x\\sinh y\\\\\\cosh(x+y)&=\\cosh x\\cosh y+\\sinh x\\sinh y\\\\\\tanh(x+y)&={\\frac {\\tanh x+\\tanh y}{1+\\tanh x\\tanh y}}\\\\\\sinh(x-y)&=\\sinh x\\cosh y-\\cosh x\\sinh y\\\\\\cosh(x-y)&=\\cosh x\\cosh y-\\sinh x\\sinh y\\\\\\tanh(x-y)&={\\frac {\\tanh x-\\tanh y}{1-\\tanh x\\tanh y}}\\\\\\end{aligned}}}  particularly cosh ( 2 x ) \= sinh 2 x \+ cosh 2 x \= 2 sinh 2 x \+ 1 \= 2 cosh 2 x − 1 sinh ( 2 x ) \= 2 sinh x cosh x tanh ( 2 x ) \= 2 tanh x 1 \+ tanh 2 x {\\displaystyle {\\begin{aligned}\\cosh(2x)&=\\sinh ^{2}{x}+\\cosh ^{2}{x}=2\\sinh ^{2}x+1=2\\cosh ^{2}x-1\\\\\\sinh(2x)&=2\\sinh x\\cosh x\\\\\\tanh(2x)&={\\frac {2\\tanh x}{1+\\tanh ^{2}x}}\\\\\\end{aligned}}} 
### Addition and subtraction formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=12 "Edit section: Addition and subtraction formulas")\]
sinh x \+ sinh y \= 2 sinh ( x \+ y 2 ) cosh ( x − y 2 ) cosh x \+ cosh y \= 2 cosh ( x \+ y 2 ) cosh ( x − y 2 ) sinh x − sinh y \= 2 cosh ( x \+ y 2 ) sinh ( x − y 2 ) cosh x − cosh y \= 2 sinh ( x \+ y 2 ) sinh ( x − y 2 ) {\\displaystyle {\\begin{aligned}\\sinh x+\\sinh y&=2\\sinh \\left({\\frac {x+y}{2}}\\right)\\cosh \\left({\\frac {x-y}{2}}\\right)\\\\\\cosh x+\\cosh y&=2\\cosh \\left({\\frac {x+y}{2}}\\right)\\cosh \\left({\\frac {x-y}{2}}\\right)\\\\\\sinh x-\\sinh y&=2\\cosh \\left({\\frac {x+y}{2}}\\right)\\sinh \\left({\\frac {x-y}{2}}\\right)\\\\\\cosh x-\\cosh y&=2\\sinh \\left({\\frac {x+y}{2}}\\right)\\sinh \\left({\\frac {x-y}{2}}\\right)\\\\\\end{aligned}}} 
### Product formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=13 "Edit section: Product formulas")\]
cosh x cosh y \= 1 2 ( cosh ( x \+ y ) \+ cosh ( x − y ) ) sinh x sinh y \= 1 2 ( cosh ( x \+ y ) − cosh ( x − y ) ) sinh x cosh y \= 1 2 ( sinh ( x \+ y ) \+ sinh ( x − y ) ) cosh x sinh y \= 1 2 ( sinh ( x \+ y ) − sinh ( x − y ) ) {\\displaystyle {\\begin{aligned}\\cosh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)+\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)-\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)+\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\cosh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)-\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\cosh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)+\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)-\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)+\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\cosh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)-\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcc6e905df736163061cf56b304ab50d7853739)
### Half argument formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=14 "Edit section: Half argument formulas")\]
sinh ( x 2 ) \= sinh x 2 ( cosh x \+ 1 ) \= sgn x cosh x − 1 2 cosh ( x 2 ) \= cosh x \+ 1 2 tanh ( x 2 ) \= sinh x cosh x \+ 1 \= sgn x cosh x − 1 cosh x \+ 1 \= e x − 1 e x \+ 1 {\\displaystyle {\\begin{aligned}\\sinh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\sqrt {2(\\cosh x+1)}}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{2}}}\\\\\[6px\]\\cosh \\left({\\frac {x}{2}}\\right)&={\\sqrt {\\frac {\\cosh x+1}{2}}}\\\\\[6px\]\\tanh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\cosh x+1}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{\\cosh x+1}}}={\\frac {e^{x}-1}{e^{x}+1}}\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}\\sinh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\sqrt {2(\\cosh x+1)}}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{2}}}\\\\\[6px\]\\cosh \\left({\\frac {x}{2}}\\right)&={\\sqrt {\\frac {\\cosh x+1}{2}}}\\\\\[6px\]\\tanh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\cosh x+1}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{\\cosh x+1}}}={\\frac {e^{x}-1}{e^{x}+1}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412a4ffd109486f684e515634b33447b13444954)
where sgn is the [sign function](https://en.wikipedia.org/wiki/Sign_function "Sign function").
If *x* ≠ 0 then
tanh ( x 2 ) \= cosh x − 1 sinh x \= coth x − csch x {\\displaystyle \\tanh \\left({\\frac {x}{2}}\\right)={\\frac {\\cosh x-1}{\\sinh x}}=\\coth x-\\operatorname {csch} x} 
### Tangent half argument formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=15 "Edit section: Tangent half argument formulas")\]
When t \= tanh ( x 2 ) {\\displaystyle t=\\tanh \\left({\\frac {x}{2}}\\right)}  , sinh x \= 2 t 1 − t 2 , cosh x \= 1 \+ t 2 1 − t 2 , tanh x \= 2 t 1 \+ t 2 , coth x \= 1 \+ t 2 2 t , sech x \= 1 − t 2 1 \+ t 2 , csch x \= 1 − t 2 2 t . {\\displaystyle {\\begin{aligned}&\\sinh x={\\frac {2t}{1-t^{2}}},&&\\cosh x={\\frac {1+t^{2}}{1-t^{2}}},\\\\\[8pt\]&\\tanh x={\\frac {2t}{1+t^{2}}},&&\\coth x={\\frac {1+t^{2}}{2t}},\\\\\[8pt\]&\\operatorname {sech} x={\\frac {1-t^{2}}{1+t^{2}}},&&\\operatorname {csch} x={\\frac {1-t^{2}}{2t}}.\\end{aligned}}} ![{\\displaystyle {\\begin{aligned}&\\sinh x={\\frac {2t}{1-t^{2}}},&&\\cosh x={\\frac {1+t^{2}}{1-t^{2}}},\\\\\[8pt\]&\\tanh x={\\frac {2t}{1+t^{2}}},&&\\coth x={\\frac {1+t^{2}}{2t}},\\\\\[8pt\]&\\operatorname {sech} x={\\frac {1-t^{2}}{1+t^{2}}},&&\\operatorname {csch} x={\\frac {1-t^{2}}{2t}}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e6413116e81cae13055fdf64801ff32f597a5)
### Square formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=16 "Edit section: Square formulas")\]
sinh 2 x \= 1 2 ( cosh 2 x − 1 ) cosh 2 x \= 1 2 ( cosh 2 x \+ 1 ) {\\displaystyle {\\begin{aligned}\\sinh ^{2}x&={\\tfrac {1}{2}}(\\cosh 2x-1)\\\\\\cosh ^{2}x&={\\tfrac {1}{2}}(\\cosh 2x+1)\\end{aligned}}} 
### Inequalities
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=17 "Edit section: Inequalities")\]
The following inequality is useful in statistics:[\[21\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-21) cosh ( t ) ≤ e t 2 / 2 . {\\displaystyle \\operatorname {cosh} (t)\\leq e^{t^{2}/2}.} 
It can be proved by comparing the Taylor series of the two functions term by term.
## Inverse functions as logarithms
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=18 "Edit section: Inverse functions as logarithms")\]
Main article: [Inverse hyperbolic function](https://en.wikipedia.org/wiki/Inverse_hyperbolic_function "Inverse hyperbolic function")
arsinh ( x ) \= ln ( x \+ x 2 \+ 1 ) arcosh ( x ) \= ln ( x \+ x 2 − 1 ) x ≥ 1 artanh ( x ) \= 1 2 ln ( 1 \+ x 1 − x ) \| x \| \< 1 arcoth ( x ) \= 1 2 ln ( x \+ 1 x − 1 ) \| x \| \> 1 arsech ( x ) \= ln ( 1 x \+ 1 x 2 − 1 ) \= ln ( 1 \+ 1 − x 2 x ) 0 \< x ≤ 1 arcsch ( x ) \= ln ( 1 x \+ 1 x 2 \+ 1 ) x ≠ 0 {\\displaystyle {\\begin{aligned}\\operatorname {arsinh} (x)&=\\ln \\left(x+{\\sqrt {x^{2}+1}}\\right)\\\\\\operatorname {arcosh} (x)&=\\ln \\left(x+{\\sqrt {x^{2}-1}}\\right)&\&x\\geq 1\\\\\\operatorname {artanh} (x)&={\\frac {1}{2}}\\ln \\left({\\frac {1+x}{1-x}}\\right)&&\|x\|\<1\\\\\\operatorname {arcoth} (x)&={\\frac {1}{2}}\\ln \\left({\\frac {x+1}{x-1}}\\right)&&\|x\|\>1\\\\\\operatorname {arsech} (x)&=\\ln \\left({\\frac {1}{x}}+{\\sqrt {{\\frac {1}{x^{2}}}-1}}\\right)=\\ln \\left({\\frac {1+{\\sqrt {1-x^{2}}}}{x}}\\right)&&0\<x\\leq 1\\\\\\operatorname {arcsch} (x)&=\\ln \\left({\\frac {1}{x}}+{\\sqrt {{\\frac {1}{x^{2}}}+1}}\\right)&\&x\\neq 0\\end{aligned}}} 
## Derivatives
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=19 "Edit section: Derivatives")\]
d d x sinh x \= cosh x d d x cosh x \= sinh x d d x tanh x \= 1 − tanh 2 x \= sech 2 x \= 1 cosh 2 x d d x coth x \= 1 − coth 2 x \= − csch 2 x \= − 1 sinh 2 x x ≠ 0 d d x sech x \= − tanh x sech x d d x csch x \= − coth x csch x x ≠ 0 {\\displaystyle {\\begin{aligned}{\\frac {d}{dx}}\\sinh x&=\\cosh x\\\\{\\frac {d}{dx}}\\cosh x&=\\sinh x\\\\{\\frac {d}{dx}}\\tanh x&=1-\\tanh ^{2}x=\\operatorname {sech} ^{2}x={\\frac {1}{\\cosh ^{2}x}}\\\\{\\frac {d}{dx}}\\coth x&=1-\\coth ^{2}x=-\\operatorname {csch} ^{2}x=-{\\frac {1}{\\sinh ^{2}x}}&\&x\\neq 0\\\\{\\frac {d}{dx}}\\operatorname {sech} x&=-\\tanh x\\operatorname {sech} x\\\\{\\frac {d}{dx}}\\operatorname {csch} x&=-\\coth x\\operatorname {csch} x&\&x\\neq 0\\end{aligned}}}  d d x arsinh x \= 1 x 2 \+ 1 d d x arcosh x \= 1 x 2 − 1 1 \< x d d x artanh x \= 1 1 − x 2 \| x \| \< 1 d d x arcoth x \= 1 1 − x 2 1 \< \| x \| d d x arsech x \= − 1 x 1 − x 2 0 \< x \< 1 d d x arcsch x \= − 1 \| x \| 1 \+ x 2 x ≠ 0 {\\displaystyle {\\begin{aligned}{\\frac {d}{dx}}\\operatorname {arsinh} x&={\\frac {1}{\\sqrt {x^{2}+1}}}\\\\{\\frac {d}{dx}}\\operatorname {arcosh} x&={\\frac {1}{\\sqrt {x^{2}-1}}}&&1\<x\\\\{\\frac {d}{dx}}\\operatorname {artanh} x&={\\frac {1}{1-x^{2}}}&&\|x\|\<1\\\\{\\frac {d}{dx}}\\operatorname {arcoth} x&={\\frac {1}{1-x^{2}}}&&1\<\|x\|\\\\{\\frac {d}{dx}}\\operatorname {arsech} x&=-{\\frac {1}{x{\\sqrt {1-x^{2}}}}}&&0\<x\<1\\\\{\\frac {d}{dx}}\\operatorname {arcsch} x&=-{\\frac {1}{\|x\|{\\sqrt {1+x^{2}}}}}&\&x\\neq 0\\end{aligned}}} 
## Second derivatives
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=20 "Edit section: Second derivatives")\]
Each of the functions sinh and cosh is equal to its [second derivative](https://en.wikipedia.org/wiki/Second_derivative "Second derivative"), that is: d 2 d x 2 sinh x \= sinh x {\\displaystyle {\\frac {d^{2}}{dx^{2}}}\\sinh x=\\sinh x}  d 2 d x 2 cosh x \= cosh x . {\\displaystyle {\\frac {d^{2}}{dx^{2}}}\\cosh x=\\cosh x\\,.} 
All functions with this property are [linear combinations](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of sinh and cosh, in particular the [exponential functions](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") e x {\\displaystyle e^{x}}  and e − x {\\displaystyle e^{-x}} .[\[22\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-22)
## Standard integrals
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=21 "Edit section: Standard integrals")\]
For a full list, see [list of integrals of hyperbolic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions "List of integrals of hyperbolic functions").
∫ sinh ( a x ) d x \= a − 1 cosh ( a x ) \+ C ∫ cosh ( a x ) d x \= a − 1 sinh ( a x ) \+ C ∫ tanh ( a x ) d x \= a − 1 ln ( cosh ( a x ) ) \+ C ∫ coth ( a x ) d x \= a − 1 ln \| sinh ( a x ) \| \+ C ∫ sech ( a x ) d x \= a − 1 arctan ( sinh ( a x ) ) \+ C ∫ csch ( a x ) d x \= a − 1 ln \| tanh ( a x 2 ) \| \+ C \= a − 1 ln \| coth ( a x ) − csch ( a x ) \| \+ C \= − a − 1 arcoth ( cosh ( a x ) ) \+ C {\\displaystyle {\\begin{aligned}\\int \\sinh(ax)\\,dx&=a^{-1}\\cosh(ax)+C\\\\\\int \\cosh(ax)\\,dx&=a^{-1}\\sinh(ax)+C\\\\\\int \\tanh(ax)\\,dx&=a^{-1}\\ln(\\cosh(ax))+C\\\\\\int \\coth(ax)\\,dx&=a^{-1}\\ln \\left\|\\sinh(ax)\\right\|+C\\\\\\int \\operatorname {sech} (ax)\\,dx&=a^{-1}\\arctan(\\sinh(ax))+C\\\\\\int \\operatorname {csch} (ax)\\,dx&=a^{-1}\\ln \\left\|\\tanh \\left({\\frac {ax}{2}}\\right)\\right\|+C=a^{-1}\\ln \\left\|\\coth \\left(ax\\right)-\\operatorname {csch} \\left(ax\\right)\\right\|+C=-a^{-1}\\operatorname {arcoth} \\left(\\cosh \\left(ax\\right)\\right)+C\\end{aligned}}} 
The following integrals can be proved using [hyperbolic substitution](https://en.wikipedia.org/wiki/Hyperbolic_substitution "Hyperbolic substitution"): ∫ 1 a 2 \+ u 2 d u \= arsinh ( u a ) \+ C ∫ 1 u 2 − a 2 d u \= sgn u arcosh \| u a \| \+ C ∫ 1 a 2 − u 2 d u \= a − 1 artanh ( u a ) \+ C u 2 \< a 2 ∫ 1 a 2 − u 2 d u \= a − 1 arcoth ( u a ) \+ C u 2 \> a 2 ∫ 1 u a 2 − u 2 d u \= − a − 1 arsech \| u a \| \+ C ∫ 1 u a 2 \+ u 2 d u \= − a − 1 arcsch \| u a \| \+ C {\\displaystyle {\\begin{aligned}\\int {{\\frac {1}{\\sqrt {a^{2}+u^{2}}}}\\,du}&=\\operatorname {arsinh} \\left({\\frac {u}{a}}\\right)+C\\\\\\int {{\\frac {1}{\\sqrt {u^{2}-a^{2}}}}\\,du}&=\\operatorname {sgn} {u}\\operatorname {arcosh} \\left\|{\\frac {u}{a}}\\right\|+C\\\\\\int {\\frac {1}{a^{2}-u^{2}}}\\,du&=a^{-1}\\operatorname {artanh} \\left({\\frac {u}{a}}\\right)+C&\&u^{2}\<a^{2}\\\\\\int {\\frac {1}{a^{2}-u^{2}}}\\,du&=a^{-1}\\operatorname {arcoth} \\left({\\frac {u}{a}}\\right)+C&\&u^{2}\>a^{2}\\\\\\int {{\\frac {1}{u{\\sqrt {a^{2}-u^{2}}}}}\\,du}&=-a^{-1}\\operatorname {arsech} \\left\|{\\frac {u}{a}}\\right\|+C\\\\\\int {{\\frac {1}{u{\\sqrt {a^{2}+u^{2}}}}}\\,du}&=-a^{-1}\\operatorname {arcsch} \\left\|{\\frac {u}{a}}\\right\|+C\\end{aligned}}} 
where *C* is the [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration").
## Taylor series expressions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=22 "Edit section: Taylor series expressions")\]
It is possible to express explicitly the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") at zero (or the [Laurent series](https://en.wikipedia.org/wiki/Laurent_series "Laurent series"), if the function is not defined at zero) of the above functions.
sinh x \= x \+ x 3 3 \! \+ x 5 5 \! \+ x 7 7 \! \+ ⋯ \= ∑ n \= 0 ∞ x 2 n \+ 1 ( 2 n \+ 1 ) \! {\\displaystyle \\sinh x=x+{\\frac {x^{3}}{3!}}+{\\frac {x^{5}}{5!}}+{\\frac {x^{7}}{7!}}+\\cdots =\\sum \_{n=0}^{\\infty }{\\frac {x^{2n+1}}{(2n+1)!}}}  This series is [convergent](https://en.wikipedia.org/wiki/Convergent_series "Convergent series") for every [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") value of x. Since the function sinh *x* is [odd](https://en.wikipedia.org/wiki/Odd_function "Odd function"), only odd exponents for *x* occur in its Taylor series.
cosh x \= 1 \+ x 2 2 \! \+ x 4 4 \! \+ x 6 6 \! \+ ⋯ \= ∑ n \= 0 ∞ x 2 n ( 2 n ) \! {\\displaystyle \\cosh x=1+{\\frac {x^{2}}{2!}}+{\\frac {x^{4}}{4!}}+{\\frac {x^{6}}{6!}}+\\cdots =\\sum \_{n=0}^{\\infty }{\\frac {x^{2n}}{(2n)!}}}  This series is [convergent](https://en.wikipedia.org/wiki/Convergent_series "Convergent series") for every [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") value of x. Since the function cosh *x* is [even](https://en.wikipedia.org/wiki/Even_function "Even function"), only even exponents for x occur in its Taylor series.
The sum of the sinh and cosh series is the [infinite series](https://en.wikipedia.org/wiki/Infinite_series "Infinite series") expression of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function").
The following series are followed by a description of a subset of their [domain of convergence](https://en.wikipedia.org/wiki/Domain_of_convergence "Domain of convergence"), where the series is convergent and its sum equals the function. tanh x \= x − x 3 3 \+ 2 x 5 15 − 17 x 7 315 \+ ⋯ \= ∑ n \= 1 ∞ 2 2 n ( 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) \! , \| x \| \< π 2 coth x \= x − 1 \+ x 3 − x 3 45 \+ 2 x 5 945 \+ ⋯ \= ∑ n \= 0 ∞ 2 2 n B 2 n x 2 n − 1 ( 2 n ) \! , 0 \< \| x \| \< π sech x \= 1 − x 2 2 \+ 5 x 4 24 − 61 x 6 720 \+ ⋯ \= ∑ n \= 0 ∞ E 2 n x 2 n ( 2 n ) \! , \| x \| \< π 2 csch x \= x − 1 − x 6 \+ 7 x 3 360 − 31 x 5 15120 \+ ⋯ \= ∑ n \= 0 ∞ 2 ( 1 − 2 2 n − 1 ) B 2 n x 2 n − 1 ( 2 n ) \! , 0 \< \| x \| \< π {\\displaystyle {\\begin{aligned}\\tanh x&=x-{\\frac {x^{3}}{3}}+{\\frac {2x^{5}}{15}}-{\\frac {17x^{7}}{315}}+\\cdots =\\sum \_{n=1}^{\\infty }{\\frac {2^{2n}(2^{2n}-1)B\_{2n}x^{2n-1}}{(2n)!}},\\qquad \\left\|x\\right\|\<{\\frac {\\pi }{2}}\\\\\\coth x&=x^{-1}+{\\frac {x}{3}}-{\\frac {x^{3}}{45}}+{\\frac {2x^{5}}{945}}+\\cdots =\\sum \_{n=0}^{\\infty }{\\frac {2^{2n}B\_{2n}x^{2n-1}}{(2n)!}},\\qquad 0\<\\left\|x\\right\|\<\\pi \\\\\\operatorname {sech} x&=1-{\\frac {x^{2}}{2}}+{\\frac {5x^{4}}{24}}-{\\frac {61x^{6}}{720}}+\\cdots =\\sum \_{n=0}^{\\infty }{\\frac {E\_{2n}x^{2n}}{(2n)!}},\\qquad \\left\|x\\right\|\<{\\frac {\\pi }{2}}\\\\\\operatorname {csch} x&=x^{-1}-{\\frac {x}{6}}+{\\frac {7x^{3}}{360}}-{\\frac {31x^{5}}{15120}}+\\cdots =\\sum \_{n=0}^{\\infty }{\\frac {2(1-2^{2n-1})B\_{2n}x^{2n-1}}{(2n)!}},\\qquad 0\<\\left\|x\\right\|\<\\pi \\end{aligned}}} 
where:
- B
n
{\\displaystyle B\_{n}}
is the *n*th [Bernoulli number](https://en.wikipedia.org/wiki/Bernoulli_number "Bernoulli number")
- E
n
{\\displaystyle E\_{n}}
is the *n*th [Euler number](https://en.wikipedia.org/wiki/Euler_number "Euler number")
## Infinite products and continued fractions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=23 "Edit section: Infinite products and continued fractions")\]
The following expansions are valid in the whole complex plane:
sinh
x
\=
x
∏
n
\=
1
∞
(
1
\+
x
2
n
2
π
2
)
\=
x
1
−
x
2
2
⋅
3
\+
x
2
−
2
⋅
3
x
2
4
⋅
5
\+
x
2
−
4
⋅
5
x
2
6
⋅
7
\+
x
2
−
⋱
{\\displaystyle \\sinh x=x\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{n^{2}\\pi ^{2}}}\\right)={\\cfrac {x}{1-{\\cfrac {x^{2}}{2\\cdot 3+x^{2}-{\\cfrac {2\\cdot 3x^{2}}{4\\cdot 5+x^{2}-{\\cfrac {4\\cdot 5x^{2}}{6\\cdot 7+x^{2}-\\ddots }}}}}}}}}
cosh
x
\=
∏
n
\=
1
∞
(
1
\+
x
2
(
n
−
1
/
2
)
2
π
2
)
\=
1
1
−
x
2
1
⋅
2
\+
x
2
−
1
⋅
2
x
2
3
⋅
4
\+
x
2
−
3
⋅
4
x
2
5
⋅
6
\+
x
2
−
⋱
{\\displaystyle \\cosh x=\\prod \_{n=1}^{\\infty }\\left(1+{\\frac {x^{2}}{(n-1/2)^{2}\\pi ^{2}}}\\right)={\\cfrac {1}{1-{\\cfrac {x^{2}}{1\\cdot 2+x^{2}-{\\cfrac {1\\cdot 2x^{2}}{3\\cdot 4+x^{2}-{\\cfrac {3\\cdot 4x^{2}}{5\\cdot 6+x^{2}-\\ddots }}}}}}}}}
tanh
x
\=
1
1
x
\+
1
3
x
\+
1
5
x
\+
1
7
x
\+
⋱
{\\displaystyle \\tanh x={\\cfrac {1}{{\\cfrac {1}{x}}+{\\cfrac {1}{{\\cfrac {3}{x}}+{\\cfrac {1}{{\\cfrac {5}{x}}+{\\cfrac {1}{{\\cfrac {7}{x}}+\\ddots }}}}}}}}}
## Comparison with circular functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=24 "Edit section: Comparison with circular functions")\]
[](https://en.wikipedia.org/wiki/File:Circular_and_hyperbolic_angle.svg)
Circle and hyperbola tangent at (1, 1) display geometry of circular functions in terms of [circular sector](https://en.wikipedia.org/wiki/Sector_of_a_circle "Sector of a circle") area u and hyperbolic functions depending on [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") area u.
The hyperbolic functions represent an expansion of [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry") beyond the [circular functions](https://en.wikipedia.org/wiki/Circular_function "Circular function"). Both types depend on an [argument](https://en.wikipedia.org/wiki/Argument_of_a_function "Argument of a function"), either [circular angle](https://en.wikipedia.org/wiki/Angle "Angle") or [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle").
Since the [area of a circular sector](https://en.wikipedia.org/wiki/Circular_sector#Area "Circular sector") with radius r and angle u (in radians) is *r*2*u*/2, it will be equal to u when *r* = √2. In the diagram, such a circle is tangent to the hyperbola *xy* = 1 at (1, 1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") with area corresponding to hyperbolic angle magnitude.
The legs of the two [right triangles](https://en.wikipedia.org/wiki/Right_triangle "Right triangle") with the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse") on the ray defining the angles are of length √2 times the circular and hyperbolic functions.
The hyperbolic angle is an [invariant measure](https://en.wikipedia.org/wiki/Invariant_measure "Invariant measure") with respect to the [squeeze mapping](https://en.wikipedia.org/wiki/Squeeze_mapping "Squeeze mapping"), just as the circular angle is invariant under rotation.[\[23\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-23)
The [Gudermannian function](https://en.wikipedia.org/wiki/Gudermannian_function "Gudermannian function") gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.
The graph of the function a cosh ( x / a ) {\\displaystyle a\\cosh(x/a)}  is the [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary"), the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
## Relationship to the exponential function
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=25 "Edit section: Relationship to the exponential function")\]
The decomposition of the exponential function in its [even and odd parts](https://en.wikipedia.org/wiki/Even%E2%80%93odd_decomposition "Even–odd decomposition") gives the identities e x \= cosh x \+ sinh x , {\\displaystyle e^{x}=\\cosh x+\\sinh x,}  and e − x \= cosh x − sinh x . {\\displaystyle e^{-x}=\\cosh x-\\sinh x.}  Combined with [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") e i x \= cos x \+ i sin x , {\\displaystyle e^{ix}=\\cos x+i\\sin x,}  this gives e x \+ i y \= ( cosh x \+ sinh x ) ( cos y \+ i sin y ) {\\displaystyle e^{x+iy}=(\\cosh x+\\sinh x)(\\cos y+i\\sin y)}  for the [general complex exponential function](https://en.wikipedia.org/wiki/General_complex_exponential_function "General complex exponential function").
Additionally, e x \= 1 \+ tanh x 1 − tanh x \= 1 \+ tanh x 2 1 − tanh x 2 {\\displaystyle e^{x}={\\sqrt {\\frac {1+\\tanh x}{1-\\tanh x}}}={\\frac {1+\\tanh {\\frac {x}{2}}}{1-\\tanh {\\frac {x}{2}}}}} 
## Hyperbolic functions for complex numbers
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=26 "Edit section: Hyperbolic functions for complex numbers")\]
| | | | | | |
|---|---|---|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_Sinh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Cosh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Tanh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Coth.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Sech.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Csch.jpg) |
| sinh ( z ) {\\displaystyle \\sinh(z)}  | | | | | |
Since the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") can be defined for any [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh *z* and cosh *z* are then [holomorphic](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function").
Relationships to ordinary trigonometric functions are given by [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") for complex numbers: e i x \= cos x \+ i sin x e − i x \= cos x − i sin x {\\displaystyle {\\begin{aligned}e^{ix}&=\\cos x+i\\sin x\\\\e^{-ix}&=\\cos x-i\\sin x\\end{aligned}}}  so: cosh ( i x ) \= 1 2 ( e i x \+ e − i x ) \= cos x sinh ( i x ) \= 1 2 ( e i x − e − i x ) \= i sin x tanh ( i x ) \= i tan x cosh ( x \+ i y ) \= cosh ( x ) cos ( y ) \+ i sinh ( x ) sin ( y ) sinh ( x \+ i y ) \= sinh ( x ) cos ( y ) \+ i cosh ( x ) sin ( y ) tanh ( x \+ i y ) \= tanh ( x ) \+ i tan ( y ) 1 \+ i tanh ( x ) tan ( y ) cosh x \= cos ( i x ) sinh x \= − i sin ( i x ) tanh x \= − i tan ( i x ) {\\displaystyle {\\begin{aligned}\\cosh(ix)&={\\frac {1}{2}}\\left(e^{ix}+e^{-ix}\\right)=\\cos x\\\\\\sinh(ix)&={\\frac {1}{2}}\\left(e^{ix}-e^{-ix}\\right)=i\\sin x\\\\\\tanh(ix)&=i\\tan x\\\\\\cosh(x+iy)&=\\cosh(x)\\cos(y)+i\\sinh(x)\\sin(y)\\\\\\sinh(x+iy)&=\\sinh(x)\\cos(y)+i\\cosh(x)\\sin(y)\\\\\\tanh(x+iy)&={\\frac {\\tanh(x)+i\\tan(y)}{1+i\\tanh(x)\\tan(y)}}\\\\\\cosh x&=\\cos(ix)\\\\\\sinh x&=-i\\sin(ix)\\\\\\tanh x&=-i\\tan(ix)\\end{aligned}}} 
Thus, hyperbolic functions are [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") with respect to the imaginary component, with period 2 π i {\\displaystyle 2\\pi i}  (π i {\\displaystyle \\pi i}  for hyperbolic tangent and cotangent).
## See also
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=27 "Edit section: See also")\]
- [e (mathematical constant)](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)")
- [Equal incircles theorem](https://en.wikipedia.org/wiki/Equal_incircles_theorem "Equal incircles theorem"), based on sinh
- [Hyperbolastic functions](https://en.wikipedia.org/wiki/Hyperbolastic_functions "Hyperbolastic functions")
- [Hyperbolic growth](https://en.wikipedia.org/wiki/Hyperbolic_growth "Hyperbolic growth")
- [Inverse hyperbolic functions](https://en.wikipedia.org/wiki/Inverse_hyperbolic_function "Inverse hyperbolic function")
- [List of integrals of hyperbolic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions "List of integrals of hyperbolic functions")
- [Poinsot's spirals](https://en.wikipedia.org/wiki/Poinsot%27s_spirals "Poinsot's spirals")
- [Sigmoid function](https://en.wikipedia.org/wiki/Sigmoid_function "Sigmoid function")
- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions")
## References
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=28 "Edit section: References")\]
1. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-3)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Hyperbolic Functions"](https://mathworld.wolfram.com/HyperbolicFunctions.html). *mathworld.wolfram.com*. Retrieved 2020-08-29.
2. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-2)**
(1999) *Collins Concise Dictionary*, 4th edition, HarperCollins, Glasgow,
[ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0 00 472257 4](https://en.wikipedia.org/wiki/Special:BookSources/0_00_472257_4 "Special:BookSources/0 00 472257 4")
, p. 1386
3. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Collins_Concise_Dictionary_p._328_3-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Collins_Concise_Dictionary_p._328_3-1) *Collins Concise Dictionary*, p. 328
4. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:2_4-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:2_4-1)
["Hyperbolic Functions"](https://www.mathsisfun.com/sets/function-hyperbolic.html). *www.mathsisfun.com*. Retrieved 2020-08-29.
5. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-5)** *Collins Concise Dictionary*, p. 1520
6. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-6)** *Collins Concise Dictionary*, p. 329
7. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-7)** [tanh](http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf)
8. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-8)** *Collins Concise Dictionary*, p. 1340
9. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-9)**
[Woodhouse, N. M. J.](https://en.wikipedia.org/wiki/N._M._J._Woodhouse "N. M. J. Woodhouse") (2003), *Special Relativity*, London: Springer, p. 71, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-85233-426-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-426-0 "Special:BookSources/978-1-85233-426-0")
10. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-10)**
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1972), [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"), New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
11. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-11)** [Some examples of using **arcsinh**](https://www.google.com/books?q=arcsinh+-library) found in [Google Books](https://en.wikipedia.org/wiki/Google_Books "Google Books").
12. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-12)**
Niven, Ivan (1985). *Irrational Numbers*. Vol. 11. Mathematical Association of America. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780883850381](https://en.wikipedia.org/wiki/Special:BookSources/9780883850381 "Special:BookSources/9780883850381")
. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [10\.4169/j.ctt5hh8zn](https://www.jstor.org/stable/10.4169/j.ctt5hh8zn).
13. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:3_13-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:3_13-1)
George F. Becker; C. E. Van Orstrand (1909). [*Hyperbolic Functions*](https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator). Universal Digital Library. The Smithsonian Institution.
14. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-3) [***e***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-4) [***f***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-5)
McMahon, James (1896). [*Hyperbolic Functions*](https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up). Osmania University, Digital Library Of India. John Wiley And Sons.
15. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-3) Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. *Euler at 300: an appreciation.* Mathematical Association of America, 2007. Page 100.
16. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-16)** Becker, Georg F. *Hyperbolic functions.* Read Books, 1931. Page xlviii.
17. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-17)**
N.P., Bali (2005). [*Golden Integral Calculus*](https://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472). Firewall Media. p. 472. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[81-7008-169-6](https://en.wikipedia.org/wiki/Special:BookSources/81-7008-169-6 "Special:BookSources/81-7008-169-6")
.
18. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-18)**
Steeb, Willi-Hans (2005). [*Nonlinear Workbook, The: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic With C++, Java And Symbolicc++ Programs*](https://books.google.com/books?id=-Qo8DQAAQBAJ) (3rd ed.). World Scientific Publishing Company. p. 281. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-981-310-648-2](https://en.wikipedia.org/wiki/Special:BookSources/978-981-310-648-2 "Special:BookSources/978-981-310-648-2")
.
[Extract of page 281 (using lambda=1)](https://books.google.com/books?id=-Qo8DQAAQBAJ&pg=PA281)
19. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-19)**
Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010). [*An Atlas of Functions: with Equator, the Atlas Function Calculator*](https://books.google.com/books?id=UrSnNeJW10YC) (2nd, illustrated ed.). Springer Science & Business Media. p. 290. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-48807-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-48807-3 "Special:BookSources/978-0-387-48807-3")
.
[Extract of page 290](https://books.google.com/books?id=UrSnNeJW10YC&pg=PA290)
20. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Osborn,_1902_20-0)**
Osborn, G. (July 1902). ["Mnemonic for hyperbolic formulae"](https://zenodo.org/record/1449741). *[The Mathematical Gazette](https://en.wikipedia.org/wiki/The_Mathematical_Gazette "The Mathematical Gazette")*. **2** (34): 189. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/3602492](https://doi.org/10.2307%2F3602492). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [3602492](https://www.jstor.org/stable/3602492). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [125866575](https://api.semanticscholar.org/CorpusID:125866575).
21. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-21)**
Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627.
[\[1\]](https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827)
22. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-22)**
[Olver, Frank W. J.](https://en.wikipedia.org/wiki/Frank_W._J._Olver "Frank W. J. Olver"); Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), ["Hyperbolic functions"](http://dlmf.nist.gov/4.34), *[NIST Handbook of Mathematical Functions](https://en.wikipedia.org/wiki/Digital_Library_of_Mathematical_Functions "Digital Library of Mathematical Functions")*, Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-19225-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-19225-5 "Special:BookSources/978-0-521-19225-5")
, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [2723248](https://mathscinet.ams.org/mathscinet-getitem?mr=2723248)
.
23. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-23)** [Haskell, Mellen W.](https://en.wikipedia.org/wiki/Mellen_W._Haskell "Mellen W. Haskell"), "On the introduction of the notion of hyperbolic functions", [Bulletin of the American Mathematical Society](https://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Society "Bulletin of the American Mathematical Society") **1**:6:155–9, [full text](https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf)
## External links
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Hyperbolic functions
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| Readable Markdown | From Wikipedia, the free encyclopedia
"Hyperbolic curve" redirects here. For the geometric curve, see [Hyperbola](https://en.wikipedia.org/wiki/Hyperbola "Hyperbola").
[](https://en.wikipedia.org/wiki/File:Sinh_cosh_tanh.svg)
In [mathematics](https://en.wikipedia.org/wiki/Mathematics "Mathematics"), **hyperbolic functions** are analogues of the ordinary [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function"), but defined using the [hyperbola](https://en.wikipedia.org/wiki/Hyperbola "Hyperbola") rather than the [circle](https://en.wikipedia.org/wiki/Circle "Circle"). Just as the points (cos *t*, sin *t*) form a [circle with a unit radius](https://en.wikipedia.org/wiki/Unit_circle "Unit circle"), the points (cosh *t*, sinh *t*) form the right half of the [unit hyperbola](https://en.wikipedia.org/wiki/Unit_hyperbola "Unit hyperbola"). Also, similarly to how the derivatives of sin(*t*) and cos(*t*) are cos(*t*) and –sin(*t*) respectively, the derivatives of sinh(*t*) and cosh(*t*) are cosh(*t*) and sinh(*t*) respectively.
Hyperbolic functions are used to express the [angle of parallelism](https://en.wikipedia.org/wiki/Angle_of_parallelism "Angle of parallelism") in [hyperbolic geometry](https://en.wikipedia.org/wiki/Hyperbolic_geometry "Hyperbolic geometry"). They are used to express [Lorentz boosts](https://en.wikipedia.org/wiki/Lorentz_boost "Lorentz boost") as [hyperbolic rotations](https://en.wikipedia.org/wiki/Hyperbolic_rotation "Hyperbolic rotation") in [special relativity](https://en.wikipedia.org/wiki/Special_relativity "Special relativity"). They also occur in the solutions of many linear [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") (such as the equation defining a [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary")), [cubic equations](https://en.wikipedia.org/wiki/Cubic_equation#Hyperbolic_solution_for_one_real_root "Cubic equation"), and [Laplace's equation](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") in [Cartesian coordinates](https://en.wikipedia.org/wiki/Cartesian_coordinates "Cartesian coordinates"). [Laplace's equations](https://en.wikipedia.org/wiki/Laplace%27s_equation "Laplace's equation") are important in many areas of [physics](https://en.wikipedia.org/wiki/Physics "Physics"), including [electromagnetic theory](https://en.wikipedia.org/wiki/Electromagnetic_theory "Electromagnetic theory"), [heat transfer](https://en.wikipedia.org/wiki/Heat_transfer "Heat transfer"), and [fluid dynamics](https://en.wikipedia.org/wiki/Fluid_dynamics "Fluid dynamics").
The basic hyperbolic functions are:[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)
- **hyperbolic sine** "sinh" (),[\[2\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-2)
- **hyperbolic cosine** "cosh" (),[\[3\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Collins_Concise_Dictionary_p._328-3)
from which are derived:[\[4\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:2-4)
- **hyperbolic tangent** "tanh" (),[\[5\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-5)
- **hyperbolic cotangent** "coth" (),[\[6\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-6)[\[7\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-7)
- **hyperbolic secant** "sech" (),[\[8\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-8)
- **hyperbolic cosecant** "csch" or "cosech" ([\[3\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Collins_Concise_Dictionary_p._328-3))
corresponding to the derived trigonometric functions.
The [inverse hyperbolic functions](https://en.wikipedia.org/wiki/Inverse_hyperbolic_functions "Inverse hyperbolic functions") are:
- **inverse hyperbolic sine** "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh")[\[9\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-9)[\[10\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-10)[\[11\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-11)
- **inverse hyperbolic cosine** "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh")
- **inverse hyperbolic tangent** "artanh" (also denoted "tanh−1", "atanh" or sometimes "arctanh")
- **inverse hyperbolic cotangent** "arcoth" (also denoted "coth−1", "acoth" or sometimes "arccoth")
- **inverse hyperbolic secant** "arsech" (also denoted "sech−1", "asech" or sometimes "arcsech")
- **inverse hyperbolic cosecant** "arcsch" (also denoted "arcosech", "csch−1", "cosech−1","acsch", "acosech", or sometimes "arccsch" or "arccosech")
[](https://en.wikipedia.org/wiki/File:Hyperbolic_functions-2.svg)
A [ray](https://en.wikipedia.org/wiki/Ray_\(geometry\) "Ray (geometry)") through the [unit hyperbola](https://en.wikipedia.org/wiki/Unit_hyperbola "Unit hyperbola") *x*2 − *y*2 = 1 at the point (cosh *a*, sinh *a*), where a is twice the area between the ray, the hyperbola, and the x\-axis. For points on the hyperbola below the x\-axis, the area is considered negative (see [animated version](https://en.wikipedia.org/wiki/File:HyperbolicAnimation.gif "File:HyperbolicAnimation.gif") with comparison with the trigonometric (circular) functions).
The hyperbolic functions take an [argument](https://en.wikipedia.org/wiki/Argument_of_a_function "Argument of a function") called a [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle"). The magnitude of a hyperbolic angle is the [area](https://en.wikipedia.org/wiki/Area "Area") of its [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") to *xy* = 1. The hyperbolic functions may be defined in terms of the [legs of a right triangle](https://en.wikipedia.org/wiki/Hyperbolic_sector#Hyperbolic_triangle "Hyperbolic sector") covering this sector.
In [complex analysis](https://en.wikipedia.org/wiki/Complex_analysis "Complex analysis"), the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are [entire functions](https://en.wikipedia.org/wiki/Entire_function "Entire function"). As a result, the other hyperbolic functions are [meromorphic](https://en.wikipedia.org/wiki/Meromorphic_function "Meromorphic function") in the whole complex plane.
By [Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem "Lindemann–Weierstrass theorem"), the hyperbolic functions have a [transcendental value](https://en.wikipedia.org/wiki/Transcendental_number "Transcendental number") for every non-zero [algebraic value](https://en.wikipedia.org/wiki/Algebraic_number "Algebraic number") of the argument.[\[12\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-12)
The first known calculation of a hyperbolic trigonometry problem is attributed to [Gerardus Mercator](https://en.wikipedia.org/wiki/Gerardus_Mercator "Gerardus Mercator") when issuing the [Mercator map projection](https://en.wikipedia.org/wiki/Mercator_projection "Mercator projection") circa 1566. It requires tabulating solutions to a [transcendental equation](https://en.wikipedia.org/wiki/Transcendental_equation "Transcendental equation") involving hyperbolic functions.[\[13\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:3-13)
The first to suggest a similarity between the sector of the circle and that of the hyperbola was [Isaac Newton](https://en.wikipedia.org/wiki/Isaac_Newton "Isaac Newton") in his 1687 [*Principia Mathematica*](https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica "Philosophiæ Naturalis Principia Mathematica").[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
[Roger Cotes](https://en.wikipedia.org/wiki/Roger_Cotes "Roger Cotes") suggested to modify the trigonometric functions using the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit")  to obtain an oblate [spheroid](https://en.wikipedia.org/wiki/Spheroid "Spheroid") from a prolate one.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
Hyperbolic functions were formally introduced in 1757 by [Vincenzo Riccati](https://en.wikipedia.org/wiki/Vincenzo_Riccati "Vincenzo Riccati").[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)[\[13\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:3-13)[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15) Riccati used *Sc.* and *Cc.* (*sinus/cosinus circulare*) to refer to circular functions and *Sh.* and *Ch.* (*sinus/cosinus hyperbolico*) to refer to hyperbolic functions.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14) As early as 1759, [Daviet de Foncenex](https://en.wikipedia.org/wiki/Fran%C3%A7ois_Daviet_de_Foncenex "François Daviet de Foncenex") showed the interchangeability of the trigonometric and hyperbolic functions using the imaginary unit and extended [de Moivre's formula](https://en.wikipedia.org/wiki/De_Moivre%27s_formula "De Moivre's formula") to hyperbolic functions.[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15)[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)
During the 1760s, [Johann Heinrich Lambert](https://en.wikipedia.org/wiki/Johann_Heinrich_Lambert "Johann Heinrich Lambert") systematized the use functions and provided exponential expressions in various publications.[\[14\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:0-14)[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15) Lambert credited Riccati for the terminology and names of the functions, but altered the abbreviations to those used today.[\[15\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:4-15)[\[16\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-16)
[](https://en.wikipedia.org/wiki/File:Cartesian_hyperbolic_rhombus.svg)
Right triangles with legs proportional to sinh and cosh
With [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle") *u*, the hyperbolic functions sinh and cosh can be defined with the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") eu.[\[1\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:1-1)[\[4\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-:2-4) In the figure  .
### Exponential definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=4 "Edit section: Exponential definitions")\]
[](https://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_sinh.svg)
sinh *x* is half the [difference](https://en.wikipedia.org/wiki/Subtraction "Subtraction") of *ex* and *e*−*x*
[](https://en.wikipedia.org/wiki/File:Hyperbolic_and_exponential;_cosh.svg)
cosh *x* is the [average](https://en.wikipedia.org/wiki/Arithmetic_mean "Arithmetic mean") of *ex* and *e*−*x*
[](https://en.wikipedia.org/wiki/File:Sinh_cosh_tanh.svg)
sinh, cosh and tanh
[](https://en.wikipedia.org/wiki/File:Csch_sech_coth.svg)
csch, sech and coth
### Differential equation definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=5 "Edit section: Differential equation definitions")\]
The hyperbolic functions may be defined as solutions of [differential equations](https://en.wikipedia.org/wiki/Differential_equation "Differential equation"): The hyperbolic sine and cosine are the solution (*s*, *c*) of the system  with the initial conditions  The initial conditions make the solution unique; without them any pair of functions  would be a solution.
sinh(*x*) and cosh(*x*) are also the unique solution of the equation *f* ″(*x*) = *f* (*x*), such that *f* (0) = 1, *f* ′(0) = 0 for the hyperbolic cosine, and *f* (0) = 0, *f* ′(0) = 1 for the hyperbolic sine.
### Complex trigonometric definitions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=6 "Edit section: Complex trigonometric definitions")\]
Hyperbolic functions may also be deduced from [trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_function "Trigonometric function") with [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") arguments:
where i is the [imaginary unit](https://en.wikipedia.org/wiki/Imaginary_unit "Imaginary unit") with *i*2 = −1.
The above definitions are related to the exponential definitions via [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") (See [§ Hyperbolic functions for complex numbers](https://en.wikipedia.org/wiki/Hyperbolic_functions#Hyperbolic_functions_for_complex_numbers) below).
## Characterizing properties
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=7 "Edit section: Characterizing properties")\]
It can be shown that the [area under the curve](https://en.wikipedia.org/wiki/Area_under_the_curve "Area under the curve") of the hyperbolic cosine (over a finite interval) is always equal to the [arc length](https://en.wikipedia.org/wiki/Arc_length "Arc length") corresponding to that interval:[\[17\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-17) 
The hyperbolic tangent is the (unique) solution to the [differential equation](https://en.wikipedia.org/wiki/Differential_equation "Differential equation") *f* ′ = 1 − *f* 2, with *f* (0) = 0.[\[18\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-18)[\[19\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-19)
The hyperbolic functions satisfy many identities, all of them similar in form to the [trigonometric identities](https://en.wikipedia.org/wiki/Trigonometric_identity "Trigonometric identity"). In fact, **Osborn's rule**[\[20\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-Osborn,_1902-20) (named after [George Osborn](https://en.wikipedia.org/wiki/George_Osborn_\(mathematician\) "George Osborn (mathematician)")) states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) for , ,  or  and  into a hyperbolic identity, by:
1. expanding it completely in terms of integral powers of sines and cosines,
2. changing sine to sinh and cosine to cosh, and
3. switching the sign of every term containing a product of two sinhs.
[Odd](https://en.wikipedia.org/wiki/Odd_function "Odd function") and [even](https://en.wikipedia.org/wiki/Even_function "Even function") functions: 
Reciprocals:
Analogous to [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula"):
Analogous to the [Pythagorean trigonometric identity](https://en.wikipedia.org/wiki/Pythagorean_trigonometric_identity "Pythagorean trigonometric identity"):
### Sums and differences of arguments
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=11 "Edit section: Sums and differences of arguments")\]
 particularly 
### Addition and subtraction formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=12 "Edit section: Addition and subtraction formulas")\]
![{\\displaystyle {\\begin{aligned}\\cosh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)+\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\cosh(x+y)-\\cosh(x-y){\\bigr )}\\\\\[5mu\]\\sinh x\\,\\cosh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)+\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\cosh x\\,\\sinh y&={\\tfrac {1}{2}}{\\bigl (}\\!\\!~\\sinh(x+y)-\\sinh(x-y){\\bigr )}\\\\\[5mu\]\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcc6e905df736163061cf56b304ab50d7853739)
### Half argument formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=14 "Edit section: Half argument formulas")\]
![{\\displaystyle {\\begin{aligned}\\sinh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\sqrt {2(\\cosh x+1)}}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{2}}}\\\\\[6px\]\\cosh \\left({\\frac {x}{2}}\\right)&={\\sqrt {\\frac {\\cosh x+1}{2}}}\\\\\[6px\]\\tanh \\left({\\frac {x}{2}}\\right)&={\\frac {\\sinh x}{\\cosh x+1}}&&=\\operatorname {sgn} x\\,{\\sqrt {\\frac {\\cosh x-1}{\\cosh x+1}}}={\\frac {e^{x}-1}{e^{x}+1}}\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/412a4ffd109486f684e515634b33447b13444954)
where sgn is the [sign function](https://en.wikipedia.org/wiki/Sign_function "Sign function").
If *x* ≠ 0 then
### Tangent half argument formulas
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=15 "Edit section: Tangent half argument formulas")\]
When , ![{\\displaystyle {\\begin{aligned}&\\sinh x={\\frac {2t}{1-t^{2}}},&&\\cosh x={\\frac {1+t^{2}}{1-t^{2}}},\\\\\[8pt\]&\\tanh x={\\frac {2t}{1+t^{2}}},&&\\coth x={\\frac {1+t^{2}}{2t}},\\\\\[8pt\]&\\operatorname {sech} x={\\frac {1-t^{2}}{1+t^{2}}},&&\\operatorname {csch} x={\\frac {1-t^{2}}{2t}}.\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c5e6413116e81cae13055fdf64801ff32f597a5)
The following inequality is useful in statistics:[\[21\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-21) 
It can be proved by comparing the Taylor series of the two functions term by term.
## Inverse functions as logarithms
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=18 "Edit section: Inverse functions as logarithms")\]
 
Each of the functions sinh and cosh is equal to its [second derivative](https://en.wikipedia.org/wiki/Second_derivative "Second derivative"), that is:  
All functions with this property are [linear combinations](https://en.wikipedia.org/wiki/Linear_combination "Linear combination") of sinh and cosh, in particular the [exponential functions](https://en.wikipedia.org/wiki/Exponential_function "Exponential function")  and .[\[22\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-22)
The following integrals can be proved using [hyperbolic substitution](https://en.wikipedia.org/wiki/Hyperbolic_substitution "Hyperbolic substitution"): 
where *C* is the [constant of integration](https://en.wikipedia.org/wiki/Constant_of_integration "Constant of integration").
## Taylor series expressions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=22 "Edit section: Taylor series expressions")\]
It is possible to express explicitly the [Taylor series](https://en.wikipedia.org/wiki/Taylor_series "Taylor series") at zero (or the [Laurent series](https://en.wikipedia.org/wiki/Laurent_series "Laurent series"), if the function is not defined at zero) of the above functions.
 This series is [convergent](https://en.wikipedia.org/wiki/Convergent_series "Convergent series") for every [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") value of x. Since the function sinh *x* is [odd](https://en.wikipedia.org/wiki/Odd_function "Odd function"), only odd exponents for *x* occur in its Taylor series.
 This series is [convergent](https://en.wikipedia.org/wiki/Convergent_series "Convergent series") for every [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") value of x. Since the function cosh *x* is [even](https://en.wikipedia.org/wiki/Even_function "Even function"), only even exponents for x occur in its Taylor series.
The sum of the sinh and cosh series is the [infinite series](https://en.wikipedia.org/wiki/Infinite_series "Infinite series") expression of the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function").
The following series are followed by a description of a subset of their [domain of convergence](https://en.wikipedia.org/wiki/Domain_of_convergence "Domain of convergence"), where the series is convergent and its sum equals the function. 
where:
## Infinite products and continued fractions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=23 "Edit section: Infinite products and continued fractions")\]
The following expansions are valid in the whole complex plane:
## Comparison with circular functions
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=24 "Edit section: Comparison with circular functions")\]
[](https://en.wikipedia.org/wiki/File:Circular_and_hyperbolic_angle.svg)
Circle and hyperbola tangent at (1, 1) display geometry of circular functions in terms of [circular sector](https://en.wikipedia.org/wiki/Sector_of_a_circle "Sector of a circle") area u and hyperbolic functions depending on [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") area u.
The hyperbolic functions represent an expansion of [trigonometry](https://en.wikipedia.org/wiki/Trigonometry "Trigonometry") beyond the [circular functions](https://en.wikipedia.org/wiki/Circular_function "Circular function"). Both types depend on an [argument](https://en.wikipedia.org/wiki/Argument_of_a_function "Argument of a function"), either [circular angle](https://en.wikipedia.org/wiki/Angle "Angle") or [hyperbolic angle](https://en.wikipedia.org/wiki/Hyperbolic_angle "Hyperbolic angle").
Since the [area of a circular sector](https://en.wikipedia.org/wiki/Circular_sector#Area "Circular sector") with radius r and angle u (in radians) is *r*2*u*/2, it will be equal to u when *r* = √2. In the diagram, such a circle is tangent to the hyperbola *xy* = 1 at (1, 1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red regions together depict a [hyperbolic sector](https://en.wikipedia.org/wiki/Hyperbolic_sector "Hyperbolic sector") with area corresponding to hyperbolic angle magnitude.
The legs of the two [right triangles](https://en.wikipedia.org/wiki/Right_triangle "Right triangle") with the [hypotenuse](https://en.wikipedia.org/wiki/Hypotenuse "Hypotenuse") on the ray defining the angles are of length √2 times the circular and hyperbolic functions.
The hyperbolic angle is an [invariant measure](https://en.wikipedia.org/wiki/Invariant_measure "Invariant measure") with respect to the [squeeze mapping](https://en.wikipedia.org/wiki/Squeeze_mapping "Squeeze mapping"), just as the circular angle is invariant under rotation.[\[23\]](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_note-23)
The [Gudermannian function](https://en.wikipedia.org/wiki/Gudermannian_function "Gudermannian function") gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers.
The graph of the function  is the [catenary](https://en.wikipedia.org/wiki/Catenary "Catenary"), the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
## Relationship to the exponential function
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=25 "Edit section: Relationship to the exponential function")\]
The decomposition of the exponential function in its [even and odd parts](https://en.wikipedia.org/wiki/Even%E2%80%93odd_decomposition "Even–odd decomposition") gives the identities  and  Combined with [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula")  this gives  for the [general complex exponential function](https://en.wikipedia.org/wiki/General_complex_exponential_function "General complex exponential function").
Additionally, 
## Hyperbolic functions for complex numbers
\[[edit](https://en.wikipedia.org/w/index.php?title=Hyperbolic_functions&action=edit§ion=26 "Edit section: Hyperbolic functions for complex numbers")\]
| | | | | | |
|---|---|---|---|---|---|
| [](https://en.wikipedia.org/wiki/File:Complex_Sinh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Cosh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Tanh.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Coth.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Sech.jpg) | [](https://en.wikipedia.org/wiki/File:Complex_Csch.jpg) |
|  | | | | | |
Since the [exponential function](https://en.wikipedia.org/wiki/Exponential_function "Exponential function") can be defined for any [complex](https://en.wikipedia.org/wiki/Complex_number "Complex number") argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions sinh *z* and cosh *z* are then [holomorphic](https://en.wikipedia.org/wiki/Holomorphic_function "Holomorphic function").
Relationships to ordinary trigonometric functions are given by [Euler's formula](https://en.wikipedia.org/wiki/Euler%27s_formula "Euler's formula") for complex numbers:  so: 
Thus, hyperbolic functions are [periodic](https://en.wikipedia.org/wiki/Periodic_function "Periodic function") with respect to the imaginary component, with period  ( for hyperbolic tangent and cotangent).
- [e (mathematical constant)](https://en.wikipedia.org/wiki/E_\(mathematical_constant\) "E (mathematical constant)")
- [Equal incircles theorem](https://en.wikipedia.org/wiki/Equal_incircles_theorem "Equal incircles theorem"), based on sinh
- [Hyperbolastic functions](https://en.wikipedia.org/wiki/Hyperbolastic_functions "Hyperbolastic functions")
- [Hyperbolic growth](https://en.wikipedia.org/wiki/Hyperbolic_growth "Hyperbolic growth")
- [Inverse hyperbolic functions](https://en.wikipedia.org/wiki/Inverse_hyperbolic_function "Inverse hyperbolic function")
- [List of integrals of hyperbolic functions](https://en.wikipedia.org/wiki/List_of_integrals_of_hyperbolic_functions "List of integrals of hyperbolic functions")
- [Poinsot's spirals](https://en.wikipedia.org/wiki/Poinsot%27s_spirals "Poinsot's spirals")
- [Sigmoid function](https://en.wikipedia.org/wiki/Sigmoid_function "Sigmoid function")
- [Trigonometric functions](https://en.wikipedia.org/wiki/Trigonometric_functions "Trigonometric functions")
1. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:1_1-3)
[Weisstein, Eric W.](https://en.wikipedia.org/wiki/Eric_W._Weisstein "Eric W. Weisstein") ["Hyperbolic Functions"](https://mathworld.wolfram.com/HyperbolicFunctions.html). *mathworld.wolfram.com*. Retrieved 2020-08-29.
2. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-2)**
(1999) *Collins Concise Dictionary*, 4th edition, HarperCollins, Glasgow, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[0 00 472257 4](https://en.wikipedia.org/wiki/Special:BookSources/0_00_472257_4 "Special:BookSources/0 00 472257 4")
, p. 1386
3. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Collins_Concise_Dictionary_p._328_3-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Collins_Concise_Dictionary_p._328_3-1) *Collins Concise Dictionary*, p. 328
4. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:2_4-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:2_4-1)
["Hyperbolic Functions"](https://www.mathsisfun.com/sets/function-hyperbolic.html). *www.mathsisfun.com*. Retrieved 2020-08-29.
5. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-5)** *Collins Concise Dictionary*, p. 1520
6. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-6)** *Collins Concise Dictionary*, p. 329
7. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-7)** [tanh](http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/hyperbolicfunctions.pdf)
8. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-8)** *Collins Concise Dictionary*, p. 1340
9. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-9)**
[Woodhouse, N. M. J.](https://en.wikipedia.org/wiki/N._M._J._Woodhouse "N. M. J. Woodhouse") (2003), *Special Relativity*, London: Springer, p. 71, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-1-85233-426-0](https://en.wikipedia.org/wiki/Special:BookSources/978-1-85233-426-0 "Special:BookSources/978-1-85233-426-0")
10. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-10)**
[Abramowitz, Milton](https://en.wikipedia.org/wiki/Milton_Abramowitz "Milton Abramowitz"); [Stegun, Irene A.](https://en.wikipedia.org/wiki/Irene_Stegun "Irene Stegun"), eds. (1972), [*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*](https://en.wikipedia.org/wiki/Abramowitz_and_Stegun "Abramowitz and Stegun"), New York: [Dover Publications](https://en.wikipedia.org/wiki/Dover_Publications "Dover Publications"), [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-486-61272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-61272-0 "Special:BookSources/978-0-486-61272-0")
11. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-11)** [Some examples of using **arcsinh**](https://www.google.com/books?q=arcsinh+-library) found in [Google Books](https://en.wikipedia.org/wiki/Google_Books "Google Books").
12. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-12)**
Niven, Ivan (1985). *Irrational Numbers*. Vol. 11. Mathematical Association of America. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[9780883850381](https://en.wikipedia.org/wiki/Special:BookSources/9780883850381 "Special:BookSources/9780883850381")
. [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [10\.4169/j.ctt5hh8zn](https://www.jstor.org/stable/10.4169/j.ctt5hh8zn).
13. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:3_13-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:3_13-1)
George F. Becker; C. E. Van Orstrand (1909). [*Hyperbolic Functions*](https://archive.org/details/hyperbolicfuncti027682mbp/page/n49/mode/2up?q=mercator). Universal Digital Library. The Smithsonian Institution.
14. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-3) [***e***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-4) [***f***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:0_14-5)
McMahon, James (1896). [*Hyperbolic Functions*](https://archive.org/details/hyperbolicfuncti031883mbp/page/n75/mode/2up). Osmania University, Digital Library Of India. John Wiley And Sons.
15. ^ [***a***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-0) [***b***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-1) [***c***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-2) [***d***](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-:4_15-3) Bradley, Robert E.; D'Antonio, Lawrence A.; Sandifer, Charles Edward. *Euler at 300: an appreciation.* Mathematical Association of America, 2007. Page 100.
16. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-16)** Becker, Georg F. *Hyperbolic functions.* Read Books, 1931. Page xlviii.
17. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-17)**
N.P., Bali (2005). [*Golden Integral Calculus*](https://books.google.com/books?id=hfi2bn2Ly4cC&pg=PA472). Firewall Media. p. 472. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[81-7008-169-6](https://en.wikipedia.org/wiki/Special:BookSources/81-7008-169-6 "Special:BookSources/81-7008-169-6")
.
18. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-18)**
Steeb, Willi-Hans (2005). [*Nonlinear Workbook, The: Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden Markov Models, Fuzzy Logic With C++, Java And Symbolicc++ Programs*](https://books.google.com/books?id=-Qo8DQAAQBAJ) (3rd ed.). World Scientific Publishing Company. p. 281. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-981-310-648-2](https://en.wikipedia.org/wiki/Special:BookSources/978-981-310-648-2 "Special:BookSources/978-981-310-648-2")
.
[Extract of page 281 (using lambda=1)](https://books.google.com/books?id=-Qo8DQAAQBAJ&pg=PA281)
19. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-19)**
Oldham, Keith B.; Myland, Jan; Spanier, Jerome (2010). [*An Atlas of Functions: with Equator, the Atlas Function Calculator*](https://books.google.com/books?id=UrSnNeJW10YC) (2nd, illustrated ed.). Springer Science & Business Media. p. 290. [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-387-48807-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-48807-3 "Special:BookSources/978-0-387-48807-3")
.
[Extract of page 290](https://books.google.com/books?id=UrSnNeJW10YC&pg=PA290)
20. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-Osborn,_1902_20-0)**
Osborn, G. (July 1902). ["Mnemonic for hyperbolic formulae"](https://zenodo.org/record/1449741). *[The Mathematical Gazette](https://en.wikipedia.org/wiki/The_Mathematical_Gazette "The Mathematical Gazette")*. **2** (34): 189. [doi](https://en.wikipedia.org/wiki/Doi_\(identifier\) "Doi (identifier)"):[10\.2307/3602492](https://doi.org/10.2307%2F3602492). [JSTOR](https://en.wikipedia.org/wiki/JSTOR_\(identifier\) "JSTOR (identifier)") [3602492](https://www.jstor.org/stable/3602492). [S2CID](https://en.wikipedia.org/wiki/S2CID_\(identifier\) "S2CID (identifier)") [125866575](https://api.semanticscholar.org/CorpusID:125866575).
21. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-21)**
Audibert, Jean-Yves (2009). "Fast learning rates in statistical inference through aggregation". The Annals of Statistics. p. 1627.
[\[1\]](https://projecteuclid.org/download/pdfview_1/euclid.aos/1245332827)
22. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-22)**
[Olver, Frank W. J.](https://en.wikipedia.org/wiki/Frank_W._J._Olver "Frank W. J. Olver"); Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), ["Hyperbolic functions"](http://dlmf.nist.gov/4.34), *[NIST Handbook of Mathematical Functions](https://en.wikipedia.org/wiki/Digital_Library_of_Mathematical_Functions "Digital Library of Mathematical Functions")*, Cambridge University Press, [ISBN](https://en.wikipedia.org/wiki/ISBN_\(identifier\) "ISBN (identifier)")
[978-0-521-19225-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-19225-5 "Special:BookSources/978-0-521-19225-5")
, [MR](https://en.wikipedia.org/wiki/MR_\(identifier\) "MR (identifier)") [2723248](https://mathscinet.ams.org/mathscinet-getitem?mr=2723248)
.
23. **[^](https://en.wikipedia.org/wiki/Hyperbolic_functions#cite_ref-23)** [Haskell, Mellen W.](https://en.wikipedia.org/wiki/Mellen_W._Haskell "Mellen W. Haskell"), "On the introduction of the notion of hyperbolic functions", [Bulletin of the American Mathematical Society](https://en.wikipedia.org/wiki/Bulletin_of_the_American_Mathematical_Society "Bulletin of the American Mathematical Society") **1**:6:155–9, [full text](https://www.ams.org/journals/bull/1895-01-06/S0002-9904-1895-00266-9/S0002-9904-1895-00266-9.pdf)
- ["Hyperbolic functions"](https://www.encyclopediaofmath.org/index.php?title=Hyperbolic_functions), *[Encyclopedia of Mathematics](https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics "Encyclopedia of Mathematics")*, [EMS Press](https://en.wikipedia.org/wiki/European_Mathematical_Society "European Mathematical Society"), 2001 \[1994\]
- [Hyperbolic functions](https://planetmath.org/hyperbolicfunctions) on [PlanetMath](https://en.wikipedia.org/wiki/PlanetMath "PlanetMath")
- [GonioLab](https://web.archive.org/web/20071006172054/http://glab.trixon.se/): Visualization of the unit circle, trigonometric and hyperbolic functions ([Java Web Start](https://en.wikipedia.org/wiki/Java_Web_Start "Java Web Start"))
- [Web-based calculator of hyperbolic functions](http://www.calctool.org/CALC/math/trigonometry/hyperbolic) |
| Shard | 152 (laksa) |
| Root Hash | 17790707453426894952 |
| Unparsed URL | org,wikipedia!en,/wiki/Hyperbolic_functions s443 |