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

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0.1

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

1.1

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

1.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2n}}{\cosh 2\pi x+1}dx=(-1{)}^n\frac{B_{2n}\left(\frac{1}{2}\right)}{2\pi }n\in \mathbb{N}$

1.3

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

1.4

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

1.5

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

1.6

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cosh \alpha x}{\cosh \pi x}\frac{1}{1+x^2}dx=4\cos \left(\frac{\alpha }{2}\right)-\pi \cos \alpha -2\sin \alpha \log \tan (\frac{\pi }{4}+\frac{\alpha }{4})-\pi <\mathrm{R}\mathrm{e}(\alpha )<\pi$

2.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{\cosh \alpha \pi x}\frac{\beta }{x^2+{\beta }^2}dx=\psi (\frac{\alpha \beta }{2}+\frac{3}{4})-\psi (\frac{\alpha \beta }{2}+\frac{1}{4})\text{Re}(\alpha ),\text{Re}(\beta )>0$

2.2

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

2.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2n}\sin 2\pi \alpha }{\cosh 2\pi x-\cos 2\pi \alpha }dx=(-1{)}^{n+1}\frac{B_{2n+1}(\alpha )}{2n+1}n\in \mathbb{N},0<\text{Re}(\alpha )<1$

2.4

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cosh ax}{\cosh 2\pi x+\cos 2\pi b}dx=\frac{\sin ab}{\sin \left(\frac{a}{2}\right)\sin 2\pi b}0<|\text{Re}(a)|<2\pi ,0<|\text{Re}(b)|<\frac{1}{2}$

2.5

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cosh \pi x\cosh ax}{\cosh 2\pi x+\cos 2\pi b}dx=\frac{\cos ab}{2\cos \left(\frac{a}{2}\right)\cos \pi b}0<|\text{Re}(a)|<\pi ,0<|\text{Re}(b)|<\frac{1}{2}$