1 Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1; .NET CLR 1.0.3705; .NET CLR 1.1.4322) /rde/contents/library/ps2img/cgi-bin/eqn2gif.cgi 157.118.3.14 1137808646 2006/01/21 10:57:26 \int_{0}^{2\pi}\log{\left(r^2-2r\cos{\theta}+1\right)}d\theta \\ =\lim_{n\to\infty}\frac{2\pi}{n}\sum_{k=0}^{n}\log{\left(r^2-2r\cos{\frac{2k\pi}{n}}+1\right)} \\ =\lim_{n\to\infty}\frac{2\pi}{n}\sum_{k=0}^{n}\log{\left(\left(r-\cos{\frac{2k\pi}{n}}\right)^{2}+\sin^{2}{\frac{2k\pi}{n}}\right)} \\ =\lim_{n\to\infty}\frac{2\pi}{n}\sum_{k=0}^{n}\log{\left|r-\left(\cos{\frac{2k\pi}{n}}+i\sin{\frac{2k\pi}{n}}\right)\right|^{2}} \\ =\lim_{n\to\infty}\frac{4\pi}{n}\log{\left|r-\left(\cos{\frac{2\times0\pi}{n}}+i\sin{\frac{2\times0\pi}{n}}\right)\right|\ldots\left|r-\left(\cos{\frac{2n\pi}{n}}+i\sin{\frac{2n\pi}{n}}\right)\right|} \\ =\lim_{n\to\infty}\frac{4\pi}{n}\log{\left|\left(r-\left(\cos{\frac{2\times0\pi}{n}}+i\sin{\frac{2\times0\pi}{n}}\right)\right)\ldots\left(r-\left(\cos{\frac{2n\pi}{n}}+i\sin{\frac{2n\pi}{n}}\right)\right)\right|} \\ =\lim_{n\to\infty}\frac{4\pi}{n}\log{\left|r^n-1\right|} \\ =\lim_{n\to\infty}\frac{4\pi}{n}\log{\left(r^n-1\right)} \\ =\lim_{n\to\infty}\frac{4\pi}{n}\log{\left(r^{n}\left(1-\frac{1}{r^{n}}\right)\right)} \\ =\lim_{n\to\infty}{4\pi}\left(\log{r}+\frac{1}{n}\log{\left(1-\frac{1}{r^{n}}\right)}\right) \\ =4\pi\log{r} 5 変換(アンチエイリアス) <IMG SRC="../cgi-cache/20060121_105225.gif"> 正常に処理されました 使用できないTeXコマンドがありました \to \infty \to \infty \to \infty \to \infty \to \infty \to \infty \to \infty \to \infty \to \infty 20060121_105225 Sat Jan 21 10:57:26 2006