記事まとめ

$\mathrm{Hom}\left(\left(G/N,*\right),\left(R,+\right)\right)\cong\left\{f\in\mathrm{Hom}\left(\left(G,*\right),\left(R,+\right)\right):N\subseteq\ker\left(f\right)\right\}$

$\displaystyle\int\left(\mathfrak{R}\left(f^\prime\right)\right)\left(z\right)\mathrm{d}z=\left(\mathfrak{R}f\right)\left(z\right)+C$
$\displaystyle\int\left(\mathfrak{I}\left(f^\prime\right)\right)\left(z\right)\mathrm{d}z=\left(\mathfrak{I}f\right)\left(z\right)+C$

$\r{\D^\times,1,\times}$

$\d\l{h}{0}{\frac{\sin h}{h}}=1$

$\d\mathrm{col}_{n,k}\s{f;a}\r{z}=\frac{1}{n}\sum_{l=0}^{n-1}\zeta_n^{-lk}f\r{\zeta_n^l\r{z-a}+a}$

$\lim\limits_{n\rightarrow\infty}\left(\displaystyle\sum_{i=1}^{n-1}\frac{1}{i}\sum_{j=i+1}^n\frac{1}{j}+\sum_{i=n+1}^{n^2-1}\frac{1}{i}\sum_{j=i+1}^{n^2}\frac{1}{j}-\sum_{i=1}^n\frac{1}{i}\sum_{j=n+1}^{n^2}\frac{1}{j}\right) $

$\lim\limits_{x\rightarrow0}\displaystyle\sum_{r=0}^{n}{}_n\mathrm{P}_r\frac{\sin\left(x+\displaystyle\frac{n+r}{2}\pi\right)}{x^{r+1}}$

$\displaystyle\int_\gamma\frac{f\left(z\right)}{z-\alpha}dz$

$\displaystyle\int_{-\infty}^\infty\frac{e^\frac{\pi t}{2}\left(2t\cos\left(e^\frac{\pi t}{2}+\frac{\pi}{4}\right)+\left(t^2+3\right)\sin\left(e^\frac{\pi t}{2}+\frac{\pi}{4}\right)\right)}{\left(t^2+1\right)\left(t^2+9\right)}\mathrm{d}t$

$\displaystyle\int_{-\infty}^\infty e^{\pi t-\frac{e^{\pi t}}{\sqrt{2}}}\frac{\left(\left(4t\right)^2+15\right)\cos\frac{e^{\pi t}}{\sqrt{2}}-8t\sin\frac{e^{\pi t}}{\sqrt{2}}}{\left(\left(4t\right)^2+15\right)^2+\left(8t\right)^2}\mathrm{d}t$

$\d\lim_{z\rightarrow-n}\frac{\psi^\r{1}\r{z}}{\Gamma\r{z}^2}$

$\displaystyle\int_0^\infty\r{\log x}e^{-x}\cos\r{\theta-\r{\tan\theta}x}\mathrm{d}x$

$\mathrm{Im}\beta\cap\mathrm{Im}\delta=\varnothing$

$\d\su{k}{1}{q-1}{\cos\frac{2p\pi k}{q}}\ln\sin\frac{\pi k}{q}\neq\gamma+\ln2q+\frac{\pi}{2}\cot\frac{\pi p}{q}-\frac{q}{p}$

$\displaystyle\sum_{n,m=0}^\infty\frac{1}{\left(n+m+1\right)^{a+1}}=\zeta\left(a\right)$

$\displaystyle\forall n\in\Z_{\geq1},\ \forall x\in\mathrm{Map}\left(\Z\cap\left[1,n\right],\R\right),\ f\left(\sum_{k=1}^nx_k\right)=\sum_{k=1}^nf\left(x_k\right)$

$\displaystyle\int_0^\frac{\pi}{2}\sin^{\frac{1}{3}}\theta\cos^{\frac{5}{3}}\theta\ln\tan\theta\mathrm{d}\theta$

$\displaystyle\int_0^1\left(\left\lfloor\frac{\alpha}{x}\right\rfloor-\alpha\left\lfloor\frac{1}{x}\right\rfloor\right)\mathrm{d}x=\alpha\ln\alpha$

$\displaystyle\int_0^1\left\lfloor\frac{1}{x}\right\rfloor^{-1}\mathrm{d}x=\zeta\left(2\right)-1$

$\displaystyle\int_0^\infty\frac{\sin sx}{e^{2\pi x}-1}\mathrm{d}x=\frac{1}{2}\left(\frac{1}{1-e^{-s}}-\frac{1}{s}-\frac{1}{2}\right)$

$\displaystyle\int_0^\infty\frac{\cos\sqrt{x}}{e^{2\pi\sqrt{x}}-1}\mathrm{d}x=1-\frac{e}{\left(e-1\right)^2}$

$\displaystyle\int_0^1\left\lfloor\frac{1}{x}\right\rfloor^{-1}\left(\left(1+s\right)x^s-sx^{-1+s}\right)\mathrm{d}x=\zeta\left(2+s\right)-1$

$\displaystyle\int_0^\infty\ln\left(1-2\frac{\cos2\theta}{x^2}+\frac{1}{x^4}\right)\mathrm{d}x=2\pi\left|\sin\theta\right|$

$\d\i{x}{0}{\infty}{\frac{\ln x}{\cosh^2x}}=\ln\frac{\pi}{4}-\gamma$

$\d\i{x}{0}{1}{\r{\frac{1}{1-x}+\frac{1}{\ln x}}}=\gamma$

$\d\i{x}{1}{\infty}{\frac{\c{x}^2}{x^{s+1}}}=\frac{1}{s-2}-\frac{\zeta\r{s}}{s}-\frac{2\zeta\r{s-1}}{s\r{s-1}}$

$\d\i{x}{0}{1}{\frac{x^n\ln^{a-1}\frac{1}{x}}{1-x}}=\Gamma\r{a}\r{\zeta\r{a}-H_{n,a}}$

$\d\i{z}{0}{1}{\i{y}{0}{1}{\i{x}{0}{1}{\f{x+y+z}}}}=1$

$\d\i{x}{-\infty}{\infty}{\frac{\sinh\alpha x}{\sinh\pi x}\cos\beta x}=\frac{\sin\alpha}{\cos\alpha+\cosh\beta}$
$\d\i{x}{-\infty}{\infty}{\frac{\cosh\alpha x}{\sinh\pi x}\sin\beta x}=\frac{\sinh\beta}{\cos\alpha+\cosh\beta}$

$\d\i{x}{0}{1}{\r{\frac{1}{\ln x}+\frac{1}{1-x}}x^{a-1}\ln^{s-1}\frac{1}{x}}=\Gamma\r{s}\r{\zeta\r{s,a}+\frac{a^{1-s}}{1-s}}$

$\displaystyle\int_0^\infty\frac{1+2\cos x+x\sin x}{1+2x\sin x+x^2}\dd x=\frac{\pi}{1+\Omega}$

$\displaystyle\int_0^1\frac{\artanh\sqrt{1-x^2}}{1-x}\dd x$