Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,sinh)

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0.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{1+x^2}\frac{x}{\sinh \pi x}dx=2\log 2-1$

1.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{x^{n-1}}{\sinh x}dx=\frac{2^n(2^n-1)|B_n|}{n}\frac{{\pi }^n}{2^{n-1}}n\in {\mathbb{Z}}^{\ge 2}$

1.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{\sinh x}dx=2\mathrm{\Gamma }(\alpha )\lambda (\alpha )\text{Re}(\alpha )>1$

1.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{{\sinh }^{2}x}dx=\frac{\mathrm{\Gamma }(\alpha )\zeta (\alpha -1)}{2^{\alpha -2}}\text{Re}(\alpha )>2$

1.4

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sinh \alpha x}{\sinh x}dx=\pi \tan \left(\frac{\alpha \pi }{2}\right)-1<\mathrm{R}\mathrm{e}(\alpha )<1$

1.5

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\sinh \alpha x}{\sinh \pi x}\frac{1}{1+x^2}dx=-\alpha \cos \alpha +2\sin \alpha \log (2\cos \frac{\alpha }{2})-\pi <\mathrm{R}\mathrm{e}(\alpha )<\pi$

2.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{x}{\sinh \alpha \pi x}\frac{1}{x^2+{\beta }^2}dx=-\frac{1}{\alpha \beta }+\psi (\frac{\alpha \beta }{2}+\frac{1}{2})-\psi \left(\frac{\alpha \beta }{2}\right)\text{Re}(\alpha ),\text{Re}(\beta )>0$