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

Aus der Formel ${\displaystyle \int _{0}^{\infty }{\frac {x^{2n}}{\cosh 2\pi x-\cos 2\pi \alpha }}\,dx=(-1)^{n+1}\,{\frac {\frac {B_{2n+1}(\alpha )}{2n+1}}{\sin 2\pi \alpha }}}$ folgt

${\displaystyle \int _{0}^{\infty }{\frac {x^{2n}}{\cosh 2\pi x+1}}\,dx=(-1)^{n+1}\,\lim _{\alpha \to 1/2}{\frac {\frac {B_{2n+1}(\alpha )}{2n+1}}{\sin 2\pi \alpha }}{\stackrel {\text{L.H.}}{\,\,\,\,=\,\,\,\,}}(-1)^{n+1}\lim _{\alpha \to 1/2}{\frac {B_{2n}(\alpha )}{2\pi \cos 2\pi \alpha }}=(-1)^{n}\,{\frac {B_{2n}\left({\frac {1}{2}}\right)}{2\pi }}}$.

Betrachte die Formel ${\displaystyle \int _{-\infty }^{\infty }{\frac {\cosh ax}{\cosh 2\pi x+\cos 2\pi b}}\,dx={\frac {\sin ab}{\sin \left({\frac {a}{2}}\right)\,\sin 2\pi b}}}$ im Grenzfall ${\displaystyle b\to 0}$:

${\displaystyle f(a):=\int _{-\infty }^{\infty }{\frac {\cosh ax}{\cosh 2\pi x+1}}\,dx={\frac {1}{\pi }}\,{\frac {\frac {a}{2}}{\sin {\frac {a}{2}}}}}$

Auf der einen Seite ist nun ${\displaystyle f^{(2n)}(a)=\int _{-\infty }^{\infty }{\frac {\cosh ax\cdot x^{2n}}{\cosh 2\pi x+1}}\,dx\,\,\Rightarrow \,\,f^{(2n)}(0)=\int _{-\infty }^{\infty }{\frac {x^{2n}}{\cosh 2\pi x+1}}\,dx}$.

Auf der anderen Seite hat die Taylorreihenentwicklung von ${\displaystyle f\,}$ folgende Form:

${\displaystyle f(a)={\frac {1}{\pi }}\sum _{n=0}^{\infty }(-1)^{n}\left({\frac {1}{2^{2n-1}}}-1\right)B_{2n}\,{\frac {a^{2n}}{(2n)!}}\,\,\Rightarrow \,\,f^{(2n)}(0)={\frac {1}{\pi }}\,(-1)^{n}\,\left({\frac {1}{2^{2n-1}}}-1\right)B_{2n}}$

Also ist ${\displaystyle \int _{-\infty }^{\infty }{\frac {x^{2n}}{\cosh 2\pi x+1}}\,dx={\frac {1}{\pi }}\,(-1)^{n}\,\left({\frac {1}{2^{2n-1}}}-1\right)B_{2n}=(-1)^{n}\,{\frac {B_{2n}\left({\frac {1}{2}}\right)}{\pi }}}$.

Für ${\displaystyle x>0\,}$ ist ${\displaystyle {\frac {1}{\cosh x}}={\frac {2}{e^{x}+e^{-x}}}=2e^{-x}{\frac {1}{1+e^{-2x}}}=2e^{-x}\sum _{k=0}^{\infty }(-1)^{k}\,e^{-2kx}=2\sum _{k=0}^{\infty }(-1)^{k}\,e^{-(2k+1)x}}$.

Also ist ${\displaystyle {\frac {x^{\alpha -1}}{\cosh x}}=2\sum _{k=0}^{\infty }(-1)^{k}\,x^{\alpha -1}\,e^{-(2k+1)x}}$ und somit ist

${\displaystyle \int _{0}^{\infty }{\frac {x^{\alpha -1}}{\cosh x}}\,dx=2\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{\infty }x^{\alpha -1}\,e^{-(2k+1)x}\,dx=2\sum _{k=0}^{\infty }(-1)^{k}\,{\frac {\Gamma (\alpha )}{(2k+1)^{\alpha }}}=2\,\Gamma (\alpha )\,\beta (\alpha )}$.

${\displaystyle y(\alpha ):=\int _{-\infty }^{\infty }{\frac {\cosh \alpha x}{\cosh \pi x}}\,{\frac {1}{1+x^{2}}}\,dx}$

${\displaystyle \Rightarrow y''(\alpha )+y(\alpha )=\int _{-\infty }^{\infty }{\frac {\cosh \alpha x}{\cosh \pi x}}\,dx=\sec \left({\frac {\alpha }{2}}\right)}$ mit ${\displaystyle y(\pi )=\int _{-\infty }^{\infty }{\frac {dx}{1+x^{2}}}=\pi }$ und ${\displaystyle y'(0)=0}$.


Ansatz (Variation der Konstante):

${\displaystyle y(x)=c(x)\sin x+d(x)\cos x\,}$

${\displaystyle y'(x)=c(x)\cos x-d(x)\sin x+\underbrace {c'(x)\sin x+d'(x)\cos x} _{=0\,{\text{(Forderung)}}}}$

${\displaystyle y''(x)=-c(x)\sin x-d(x)\cos x+c'(x)\cos x-d'(x)\sin x\,}$

Also ist ${\displaystyle y''(x)+y(x)=c'(x)\cos x-d'(x)\sin x=\sec {\frac {x}{2}}}$.

${\displaystyle {\begin{pmatrix}\cos x&-\sin x\\\sin x&\cos x\end{pmatrix}}{\begin{pmatrix}c'(x)\\d'(x)\end{pmatrix}}={\begin{pmatrix}\sec {\frac {x}{2}}\\0\end{pmatrix}}}$

${\displaystyle \Rightarrow {\begin{pmatrix}c'(x)\\d'(x)\end{pmatrix}}={\begin{pmatrix}\cos x&\sin x\\-\sin x&\cos x\end{pmatrix}}{\begin{pmatrix}\sec {\frac {x}{2}}\\0\end{pmatrix}}={\begin{pmatrix}\cos x\,\,\sec {\frac {x}{2}}\\-\sin x\,\,\sec {\frac {x}{2}}\end{pmatrix}}={\begin{pmatrix}\left(2\cos ^{2}{\frac {x}{2}}-1\right)\sec {\frac {x}{2}}\\-2\sin {\frac {x}{2}}\cos {\frac {x}{2}}\,\sec {\frac {x}{2}}\end{pmatrix}}={\begin{pmatrix}2\cos {\frac {x}{2}}-\sec {\frac {x}{2}}\\-2\sin {\frac {x}{2}}\end{pmatrix}}}$

${\displaystyle \Rightarrow {\begin{pmatrix}c(x)\\d(x)\end{pmatrix}}={\begin{pmatrix}4\sin {\frac {x}{2}}-2\log \tan \left({\frac {\pi }{4}}+{\frac {x}{4}}\right)\\4\cos {\frac {x}{2}}-\pi \end{pmatrix}}}$, wegen ${\displaystyle c(0)=y'(0)=0}$ und ${\displaystyle -d(\pi )=y(\pi )=\pi }$.

Somit ist ${\displaystyle y(x)=\left[4\sin {\frac {x}{2}}-2\log \tan \left({\frac {\pi }{4}}+{\frac {x}{4}}\right)\right]\sin x+\left[4\cos {\frac {x}{2}}-\pi \right]\cos x}$

${\displaystyle =4\sin {\frac {x}{2}}\,\sin x-2\sin x\,\log \tan \left({\frac {\pi }{4}}+{\frac {x}{4}}\right)+4\cos {\frac {x}{2}}\,\cos x-\pi \,\cos x}$

${\displaystyle =4\cos \left({\frac {x}{2}}\right)-\,\pi \cos x-\,2\sin x\,\,\log \tan \left({\frac {\pi }{4}}+{\frac {x}{4}}\right)}$.

${\displaystyle I:=\int _{-\infty }^{\infty }{\frac {2\cosh \beta x}{(2\cosh x)^{\alpha }}}\,dx=\int _{-\infty }^{\infty }{\frac {e^{\beta x}+e^{-\beta x}}{(e^{x}+e^{-x})^{\alpha }}}\,dx=\int _{0}^{\infty }{\frac {x^{\beta }+x^{-\beta }}{(x+x^{-1})^{\alpha }}}\,{\frac {dx}{x}}}$

ist nach Substitution ${\displaystyle x\mapsto {\frac {\sin u}{\cos u}}}$ gleich ${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {\left({\frac {\sin u}{\cos u}}\right)^{\beta }+\left({\frac {\cos u}{\sin u}}\right)^{\beta }}{\left({\frac {\sin u}{\cos u}}+{\frac {\cos u}{\sin u}}\right)^{\alpha }}}\,{\frac {du}{\sin u\cdot \cos u}}}$.

Dabei ist ${\displaystyle {\frac {\sin u}{\cos u}}+{\frac {\cos u}{\sin u}}={\frac {\sin ^{2}u+\cos ^{2}u}{\sin u\cdot \cos u}}={\frac {1}{\sin u\cdot \cos u}}}$.

Also ist ${\displaystyle I=\int _{0}^{\frac {\pi }{2}}\left(\sin u\cdot \cos u\right)^{\alpha -1}\left(\left({\frac {\sin u}{\cos u}}\right)^{\beta }+\left({\frac {\cos u}{\sin u}}\right)^{\beta }\right)du}$

${\displaystyle \int _{0}^{\frac {\pi }{2}}(\sin u)^{\alpha +\beta -1}\,(\cos u)^{\alpha -\beta -1}\,du+\int _{0}^{\frac {\pi }{2}}(\sin u)^{\alpha -\beta -1}\,(\cos u)^{\alpha +\beta -1}\,du}$

${\displaystyle =B\left({\frac {\alpha +\beta }{2}},{\frac {\alpha -\beta }{2}}\right)={\frac {\Gamma \left({\frac {\alpha +\beta }{2}}\right)\,\Gamma \left({\frac {\alpha -\beta }{2}}\right)}{\Gamma (\alpha )}}}$.

Setze ${\displaystyle F_{n}(x):={\frac {1}{i^{n}}}\,{\frac {\pi }{2}}\,\int _{-\infty }^{\infty }y^{n}\,{\frac {1}{\sin ^{2}\pi (x+iy)}}\,dy}$.

${\displaystyle F_{0}(x)={\frac {\pi }{2}}\int _{-\infty }^{\infty }{\frac {1}{\sin ^{2}\pi (x+iy)}}\,dy=\left[{\frac {i}{2}}\,\cot \pi (x+iy)\right]_{y=-\infty }^{y=\infty }={\frac {1}{2}}+{\frac {1}{2}}=1}$

${\displaystyle F_{n}(0)=-{\frac {1}{i^{n}}}\,{\frac {\pi }{2}}\,\int _{-\infty }^{\infty }{\frac {y^{n}}{\sinh ^{2}\pi y}}=B_{n}\qquad n\geq 2}$

${\displaystyle F_{n}\left({\frac {1}{2}}\right)={\frac {1}{i^{n}}}\,{\frac {\pi }{2}}\,\int _{-\infty }^{\infty }{\frac {y^{n}}{\cosh ^{2}\pi y}}\,dy=\left({\frac {1}{2^{n-1}}}-1\right)B_{n}\qquad n\geq 0}$

${\displaystyle F_{n}'(x)=-{\frac {1}{i^{n}}}\,\pi ^{2}\,\int _{-\infty }^{\infty }y^{n}\,{\frac {\cos \pi (x+iy)}{\sin ^{3}\pi (x+iy)}}\,dy}$

${\displaystyle =\underbrace {\left[-{\frac {1}{i^{n-1}}}\,{\frac {\pi }{2}}\,y^{n}\,{\frac {1}{\sin ^{2}\pi (x+iy)}}\right]_{y=-\infty }^{y=\infty }} _{=0}+{\frac {1}{i^{n-1}}}\,{\frac {\pi }{2}}\,\int _{-\infty }^{\infty }n\,y^{n-1}\,{\frac {1}{\sin ^{2}\pi (x+iy)}}\,dy=n\,F_{n-1}(x)}$

Daher muss ${\displaystyle F_{n}(x)}$ das Bernoulli-Polynom ${\displaystyle B_{n}(x)}$ sein.

${\displaystyle B_{2n+1}(x)={\frac {1}{i^{2n+1}}}\,{\frac {\pi }{2}}\,\int _{-\infty }^{\infty }y^{2n+1}\,{\frac {1}{\sin ^{2}\pi (x+iy)}}\,dy}$

${\displaystyle =\lim _{M\to \infty }\left(\left[{\frac {1}{i^{2n}}}\,{\frac {1}{2}}\,y^{2n+1}\,\cot \pi (x+iy)\right]_{y=-M}^{y=M}-{\frac {1}{i^{2n}}}\,{\frac {1}{2}}\,\int _{-M}^{M}(2n+1)\,y^{2n}\,\cot \pi (x+iy)\,dy\right)}$

Wegen ${\displaystyle \cot \pi (x+iy)+\cot \pi (x-iy)=2\,{\frac {\sin 2\pi x}{\cosh 2\pi y-\cos 2\pi x}}}$ ist

${\displaystyle \lim _{M\to \infty }M^{2n+1}{\Big (}\cot \pi (x+iM)+\cot \pi (x-iM){\Big )}=0}$ und daher ist

${\displaystyle 2\,(-1)^{n+1}\,{\frac {B_{2n+1}(x)}{2n+1}}={\text{p.V.}}\int _{-\infty }^{\infty }y^{2n}\,\cot \pi (x+iy)\,dy={\text{p.V.}}\int _{-\infty }^{\infty }y^{2n}\,\cot \pi (x-iy)\,dy}$.

Also ist ${\displaystyle \int _{-\infty }^{\infty }y^{2n}\,{\frac {\sin 2\pi x}{\cosh 2\pi y-\cos 2\pi x}}\,dy}$

${\displaystyle ={\frac {1}{2}}\int _{-\infty }^{\infty }y^{2n}{\Big (}\cot \pi (x+iy)+\cot \pi (x-iy){\Big )}dy=2\,(-1)^{n+1}\,{\frac {B_{2n+1}(x)}{2n+1}}}$.

Setzt man ${\displaystyle f(z)={\frac {\cosh az}{\cosh 2\pi z+\cos 2\pi b}}}$, so ist ${\displaystyle f(z+i)={\frac {\cosh az\,\cos a+i\sinh az\,\sin a}{\cosh 2\pi z+\cos 2\pi b}}}$.

Da der Imaginärteil eine ungerade Funktion ist, gilt ${\displaystyle \int _{-\infty }^{\infty }f(x+i)\,dx=\cos a\,\int _{-\infty }^{\infty }f(x)\,dx}$.

Für ${\displaystyle R\to \infty \,}$ verschwinden die Integrale über den vertikalen Strecken, daher ist

${\displaystyle \lim _{R\to \infty }\oint f\,dz=\int _{-\infty }^{\infty }f(x)\,dx-\int _{-\infty }^{\infty }f(x+i)\,dx=(1-\cos a)\int _{-\infty }^{\infty }f\,dx=2\,\sin ^{2}\left({\frac {a}{2}}\right)\,\int _{-\infty }^{\infty }f\,dx}$.

Auf der anderen Seite ist nach dem Residuensatz ${\displaystyle \lim _{R\to \infty }\oint f\,dz=2\pi i\,{\text{res}}\left(f,i\left({\frac {1}{2}}-b\right)\right)+2\pi i\,{\text{res}}\left(f,i\left({\frac {1}{2}}+b\right)\right)}$

${\displaystyle ={\frac {\cos \left(a\left({\frac {1}{2}}-b\right)\right)}{\sin 2\pi b}}-{\frac {\cos \left(a\left({\frac {1}{2}}+b\right)\right)}{\sin 2\pi b}}={\frac {2\,\sin \left({\frac {a}{2}}\right)\,\sin ab}{\sin 2\pi b}}}$.

Daraus folgt ${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx={\frac {\sin ab}{\sin \left({\frac {a}{2}}\right)\,\sin 2\pi b}}}$.

Setzt man ${\displaystyle f(z)={\frac {\cosh \pi z\,\cosh az}{\cosh 2\pi z+\cos 2\pi b}}}$, so ist ${\displaystyle f(z+i)=-{\frac {\cosh \pi z\,(\cosh az\,\cos a+i\sinh az\,\sin a)}{\cosh 2\pi z+\cos 2\pi b}}}$.

Da der Imaginärteil eine ungerade Funktion ist, gilt ${\displaystyle \int _{-\infty }^{\infty }f(x+i)\,dx=-\cos a\,\int _{-\infty }^{\infty }f(x)\,dx}$.

Für ${\displaystyle R\to \infty \,}$ verschwinden die Integrale über den vertikalen Strecken, daher ist

${\displaystyle \lim _{R\to \infty }\oint f\,dz=\int _{-\infty }^{\infty }f(x)\,dx-\int _{-\infty }^{\infty }f(x+i)\,dx=(1+\cos a)\int _{-\infty }^{\infty }f\,dx=2\,\cos ^{2}\left({\frac {a}{2}}\right)\,\int _{-\infty }^{\infty }f\,dx}$.

Auf der anderen Seite ist nach dem Residuensatz ${\displaystyle \lim _{R\to \infty }\oint f\,dz=2\pi i\,{\text{res}}\left(f,i\left({\frac {1}{2}}-b\right)\right)+2\pi i\,{\text{res}}\left(f,i\left({\frac {1}{2}}+b\right)\right)}$

${\displaystyle ={\frac {\cos \left(a\left({\frac {1}{2}}-b\right)\right)\sin b\pi }{\sin 2\pi b}}-{\frac {\cos \left(a\left({\frac {1}{2}}+b\right)\right)\sin b\pi }{\sin 2\pi b}}=2\cos \left({\frac {a}{2}}\right)\cos ab\,{\frac {\sin b\pi }{2\sin b\pi \,\cos b\pi }}={\frac {\cos \left({\frac {a}{2}}\right)\,\cos ab}{\cos \pi b}}}$.

Daraus folgt ${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx={\frac {\cos ab}{2\,\cos \left({\frac {a}{2}}\right)\,\cos \pi b}}}$.