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| Property | Value |
|---|---|
| URL | https://mathlog.info/articles/eD5JeKj6fR5nUna80ejj |
| Last Crawled | 2026-05-24 07:08:05 (10 days ago) |
| First Indexed | 2023-10-11 01:22:07 (2 years ago) |
| HTTP Status Code | 200 |
| Content | |
| Meta Title | logใใใ็ฉๅใฎ่งฃๆณใพใจใ | Mathlog |
| Meta Description | # $\log$ใใใ็ฉๅใฎ่งฃๆณใซใคใใฆ ใฉใใใใใใใงใใ ไปๅใฏ$\log$ใใใ็ฉๅใฎ่งฃๆณใซใคใใฆใพใจใใฆใฟใพใใใ ใใใใ็ฅใฃใฆใ$\log$ใใใ็ฉๅใฏใใใใ็ฅใ้ใใฏ5ใคใใฃใฆใ้จๅ็ฉๅ,็ฝฎๆ็ฉๅ,็ดๆฐๅฑ้,ๅพฎๅ,็ๆฐๅฎ็ใฎ5ใคใงใใ ใใใใๆธใใฆใใใพใใ ## ้จๅ็ฉๅ $\displaystyle\frac{d}{dx}\log x=\frac{1}{x}$ใจ้จๅ็ฉๅใไฝฟใฃใฆ็ฉๅใ่งฃใใใใจใใใใพใใ $$\int f(x)g(x)dx=f(x)G(x)-\int f'(x)G(x)dx$$ $\log x$ใ$f$ใฎๆนใซใใใฐ$\log x$ใ$\displaystyle\frac{1}{x}$ใซใชใฃใฆ่งฃใใใใจใใใใพใใ &&& \begin{align} I&=\int_{0}^{1}x\log x dx \\&=\left[\frac{x^2}{2}\log x\right]^{1}_{0}-\frac{1}{2}\int_{0}^{1}xdx \\&=\frac{1}{2}\lim_{x\to0}x^2\log x-\frac{1}4 \\&=-\frac{1}{4} \end{align} \begin{align} \lim_{x\to0}x^2\log x &=\lim_{x\to0}\frac{\log x}{\frac{1}{x^2}} \\&=\lim_{x\to0}\frac{\frac{1}{x}}{-\frac{2}{x^3 }} \\&=-\frac{1}{2}\lim_{x\to0}x^2 \\&=0 \end{align} &&& ใใใชๆใใง้จๅ็ฉๅใง่งฃใใใใจใใใฃใใใใพใใ ๆฅต้ใงใ$\log x$ใฎๅพฎๅใ$\displaystyle\frac{1}{x}$ใงใใใใจใไฝฟใฃใฆใพใใญใ ## ็ฝฎๆ็ฉๅ $\log x=t$ใง็ฝฎๆใใใใจใง่งฃใใใใจใใใใพใใ ้้ขๆฐใฏ$e^t=x$ใงใใ $dx$ใจ$dt$ใฎ้ขไฟใฏ$\displaystyle dt=\frac{dx}{x}$ใงใใ ใใใใฏ$\displaystyle dx=e^t\ dt$ใงใใ &&& \begin{align} I&=\int_{1}^{e}\frac{\log x}{x}dx \\&=\int_{0}^{1}t\ dt\qquad(\log x\mapsto t) \\&=1 \end{align} &&& ใจใใใใๅๆฏใซ$x$ใใใใฐ$\log x=t$ใง็ฝฎๆใใฆใใใจๆใใพใใ ใใจใฏ$\mathrm{King Property}$ใไฝฟใๆนๆณใใใฃใใใใพใใ [ใใฎ่จไบ](https://mathlog.info/articles/6ogbETh0rF02owJLJzbT)ใง$\mathrm{King Property}$ใไฝฟใ็ฉๅใ่งฃใใฆใใพใใ ## ็ดๆฐๅฑ้ ๆธใใใใฎใฏใใใใใงใใ ็ดๆฐๅฑ้ใใใฆ็ฉๅใจ็ดๆฐใไบคๆใใฆ็ฉๅใ่งฃใใใใจใใใใพใใ ใใใใ็ฅใฃใฆใใ$\log$้ข้ฃใฎ็ดๆฐๅฑ้ใใพใจใใฆใฟใพใใ &&& $$\log(1-x)=-\sum_{n=1}^{\infty}\frac{x^n}{n}\quad(|x|<1)$$ $$\log(1+x)=-\sum_{n=1}^{\infty}\frac{(-1)^n\ x^n}{n}\quad(|x|<1)$$ $$\log\left(2\sin\frac{x}{2}\right)=-\sum_{n=1}^{\infty}\frac{\cos nx}{n}\quad\left(0<x<\frac{\pi}{2}\right)$$ $$\log\left(2\cos\frac{x}{2}\right)=-\sum_{n=1}^{\infty}\frac{(-1)^n\ \cos nx}{n}\quad\left(0<x<\frac{\pi}{2}\right)$$ $$\log\tan\frac{x}{2}=-2\sum_{n=1}^{\infty}\frac{\cos(2n+1)x}{2n+1}\quad\left(0<x<\frac{\pi}{2}\right)$$ &&& ็ดๆฐๅฑ้ใไฝฟใฃใฆ่งฃใใฆ่ฆใพใใใใ &&& \begin{align} I&=\int_{0}^{1}\frac{\log(1+x)}{\sqrt{x}}dx \\&=-\int_{0}^{1}\sum_{n=1}^{\infty}\frac{(-1)^n\ x^n}{n\ \sqrt{x}}dx \\&=-\sum_{n=1}^{\infty}\frac{(-1)^n}{n}\int_{0}^{1}x^{n-\frac{1}{2}}\ dx \\&=-2\sum_{n=1}^{\infty}\frac{(-1)^n}{n(2n+1)} \\&=-2\left(\sum_{n=1}^{\infty}\frac{(-1)^n}{n}-2\sum_{n=1}^{\infty}\frac{(-1)^n}{2n+1}\right) \\&=-2\left(\log2+\frac{\pi}{2}-2\right) \\&=-2\log2-\pi+4 \end{align} &&& ๆฎ้ใซ้จๅ็ฉๅใงใ่งฃใใพใใ &&& \begin{align} I&=\int_{0}^{\frac{\pi}{2}}\log\left(2\sin\frac{x}{2}\right)\log\left(2\cos\frac{x}{2}\right)dx \\&=\frac{1}{2}\int_{0}^{\pi}\log\left(2\sin\frac{x}{2}\right)\log\left(2\cos\frac{x}{2}\right)dx \\&=\frac{1}{2}\sum_{n,m>0}\frac{(-1)^n}{nm}\int_{0}^{\pi}\cos mx\cos nx dx \\&=\frac{\pi}{4}\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2} \\&=-\frac{\pi^3}{48} \end{align} &&& ้ใงใใ่จฑใใฆใใ ใใใ [ๅใฎ่จไบ](https://mathlog.info/articles/nFkkmCBNuFHtPOyrFUSk)ใงๅฅใฎๆนๆณใง่งฃใใฆใพใใ ## ๅพฎๅ $\displaystyle\frac{d}{ds}x^s=x^s\log x$ใไฝฟใฃใฆ็ฉๅใ่งฃใใใใจใใใใพใใ &&& \begin{align} I&=\int_{0}^{\infty}e^{-x}\log xdx \\&=\left.\frac{d}{ds}\int_{0}^{\infty}x^s e^{-x}dx\right|_{s=1} \\&=\Gamma'(1) \\&=\Gamma(1)\psi(1) \\&=-\gamma \end{align} &&& ๅพฎๅใจ็ฉๅใฎไบคๆใไฝฟใใพใใใ &&& $$I=\int_{0}^{1}\frac{x^2-x}{\log x}dx$$ $$f(t)=\int_{0}^{1}\frac{x^t}{\log x}$$ \begin{align} f'(t)&=\int_{0}^{1}x^t \\&=\frac{1}{t+1} \end{align} \begin{align} f(2)-f(1)&=\int_{1}^{2}f'(t)dt \\&=\int_{1}^{2}\frac{1}{t+1}dt \\&=\log\frac{3}{2} \end{align} &&& ใใฎ่งฃๆณใใใชใๅฅฝใใงใใ ## ่ค็ด ็ฉๅ ่ค็ด ็ฉๅใง่งฃใใใใจใใใฃใใใใใ ใใใใๆธใใฆใ่ค็ด ็ฉๅใฎ2ใคใฎ่จไบใฏใฉใฃใกใ$\log$ใใใใฎใงใใกใใใฟใฆใปใใใ([ใใ](https://mathlog.info/articles/dGB4ngusgBuBJM1K6rG6)ใจ[ใใ](https://mathlog.info/articles/uqiFFCC5kXHH58DYZkks)) ่ขซ็ฉๅ้ขๆฐใซ$\log x$ใ ใฃใใใใผใ ใฏใผใใณใฎใใใช็ต่ทฏใง่งฃใใฆใใใใจใๅคใๆฐใใใพใใ ใใผใ ใฏใผใใณใฎใใใช็ต่ทฏใฏๅใฎ่จไบใงใใฃใฆใใใ &&& $$I=\int_{0}^{\infty}\frac{x\log x}{x^4+1}dx$$ $$f(z)=\frac{z\log z}{z^4+1}$$  $\log0$ใๅ้ฟใใใใใซ$0$ใฏ้ฟใใฆใใพใใ $\displaystyle C_2,C_4$ใฏ่ฉไพกใใฆๆฅต้ใจใฐใใจ$0$ใซใชใใพใใ ็ขบ่ชใใฆใฟใฆใใ ใใใ ๅ จไฝใฎ็ฉๅใ$C$ใจใใฆใใใพใใ ใใใใ่จ็ฎใใฆใใใพใใ \begin{align} \int_{C_1}&=\int_{\varepsilon}^{R}f(z)dz \\&=I\qquad(\ip\to0,R\to\infty) \end{align} \begin{align} \int_{C_3}&=\int_{iR}^{i\ip}f(z)dz \\&=-i\int_{\ip}^{R}f(ix)dx \\&=\int_{\ip}^{R}\frac{x\log ix}{x^4+1}dx \\&=\int_{\ip}^{R}\frac{x\log x}{x^4+1}dx+\frac{i\pi}{2}\int_{\ip}^{R}\frac{x}{x^4+1}dx \\&=I+\frac{\pi^2}{8}i\qquad(\ip\to0,R\to\infty) \end{align} \begin{align} \oint_{C}&=2\pi i\underset{z=e^{\frac{i\pi}{4}}}{\mathrm{Res}}\frac{z\log z}{z^4+1} \\&=2\pi i\lim_{z\to e^{\frac{i\pi}{4}}}\frac{z\log z(z-e^{\frac{i\pi}{4}})}{z^4+1} \\&=2\pi i\lim_{z\to e^{\frac{i\pi}{4}}}\frac{(2z-e^{\frac{i\pi}{4}})\log z-e^{\frac{i\pi}{4}}+z}{4z^3} \\&=\frac{\pi^2}{8}i \end{align} $$\oint_{C}=\int_{C_{1}}+\int_{C_{2}}+\int_{C_{3}}+\int_{C_{4}}$$ $$\frac{\pi^2}{8}i=I+I+\frac{\pi^2}{8}i$$ $$I=0$$ &&& ๅพฎๅใไฝฟใฃใฆใ่งฃใใพใใ ่ขซ็ฉๅ้ขๆฐใฎ$\log x$ใฎๆๆฐใซ$1$ใ่ถณใใฆ$f(z)$ใ่จญๅฎใใใใจใใใใพใใ ใใใพใผใ |
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| Language | ja |
| Author | null |
| Publish Time | not set |
| Original Publish Time | 2023-10-11 01:22:07 (2 years ago) |
| Republished | No |
| Word Count (Total) | 292 |
| Word Count (Content) | 245 |
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| External Links | 1 |
| Internal Links | 11 |
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| Performance | |
| Download Time (ms) | 357 |
| TTFB (ms) | 356 |
| Download Size (bytes) | 16,364 |
| Location | |
| Host ID | 46 (laksa046) |
| Partition ID | 42 |
| Root Hash | 792424158193528446 |
| Unparsed URL | info,mathlog!/articles/eD5JeKj6fR5nUna80ejj s443 |