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

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3.1

$\int _{-\infty }^{\infty }{\frac {\cos ax}{x}}\,{\frac {\sinh \beta x}{\cosh \gamma x}}\,dx=\log {\frac {\cosh {\frac {a\pi }{2\gamma }}+\sin {\frac {\beta \pi }{2\gamma }}}{\cosh {\frac {a\pi }{2\gamma }}-\sin {\frac {\beta \pi }{2\gamma }}}}\qquad |{\text{Re}}(\beta )|+|{\text{Im}}(a)|<{\text{Re}}(\gamma )$

Beweis

Betrachte die Formel $\int _{-\infty }^{\infty }{\frac {\cosh ux}{\cosh \gamma x}}\,dx={\frac {\pi }{\gamma }}\,\sec \left({\frac {u\pi }{2\gamma }}\right)$ für $u=\beta \pm ia$ mit $\left|{\text{Re}}(\beta )\right|+\left|{\text{Im}}(a)\right|<{\text{Re}}(\gamma )$.

Wegen $\cosh(\beta +ia)x+\cosh(\beta -ia)x=2\cosh \beta x\,\cos ax$ ist

$\int _{-\infty }^{\infty }{\frac {\cosh \beta x\,\cos ax}{\cosh \gamma x}}\,dx={\frac {\pi }{2\gamma }}\,\sec \left({\frac {(\beta +ia)\pi }{2\gamma }}\right)+{\frac {\pi }{2\gamma }}\,\sec \left({\frac {(\beta -ia)\pi }{2\gamma }}\right)$.

Integriere nach $\beta \,$:

$\int _{-\infty }^{\infty }{\frac {\sinh \beta x}{x}}\,{\frac {\cos ax}{\cosh \gamma x}}\,dx=\log \tan \left({\frac {\pi }{4}}+{\frac {\pi (\beta +ia)}{4\gamma }}\right)+\log \tan \left({\frac {\pi }{4}}+{\frac {\pi (\beta -ia)}{4\gamma }}\right)$

Und das lässt sich schreiben als

$\log \left(\tan \left({\frac {{\frac {\pi }{2}}+{\frac {\beta \pi }{2\gamma }}+{\frac {ia\pi }{2\gamma }}}{2}}\right)\cdot \tan \left({\frac {{\frac {\pi }{2}}+{\frac {\beta \pi }{2\gamma }}-{\frac {ia\pi }{2\gamma }}}{2}}\right)\right)=\log {\frac {\cos \left({\frac {ia\pi }{2\gamma }}\right)-\cos \left({\frac {\pi }{2}}+{\frac {\beta \pi }{2\gamma }}\right)}{\cos \left({\frac {ia\pi }{2\gamma }}\right)+\cos \left({\frac {\pi }{2}}+{\frac {\beta \pi }{2\gamma }}\right)}}=\log {\frac {\cosh {\frac {a\pi }{2\gamma }}+\sin {\frac {\beta \pi }{2\gamma }}}{\cosh {\frac {a\pi }{2\gamma }}-\sin {\frac {\beta \pi }{2\gamma }}}}$.