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

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2.1

$\int _{0}^{\infty }J_{0}(ax)\,{\frac {1-\cos(cx)}{x^{2}}}\,dx=\left\{{\begin{matrix}{\frac {c\pi }{2}}-a&a<c\\\\c\cdot \arcsin \left({\frac {c}{a}}\right)+{\sqrt {a^{2}-c^{2}}}-a&a>c\end{matrix}}\right.$

ohne Beweis

 

2.2

$\int _{0}^{\infty }J_{0}(ax)\,{\frac {1-\cos(cx)}{x}}\,dx=\left\{{\begin{matrix}{\text{arcosh}}\left({\frac {c}{a}}\right)&a<c\\\\0&a>c\end{matrix}}\right.$

ohne Beweis

 

2.3

$\int _{0}^{\infty }J_{0}(ax)\,\cos(cx)\,dx=\left\{{\begin{matrix}0&a<c\\\\{\frac {1}{\sqrt {a^{2}-c^{2}}}}&a>c\end{matrix}}\right.$

ohne Beweis (Weber)

 

3.1

$\int _{0}^{\infty }J_{\nu }(ax)\,{\frac {\cos(cx)}{x}}\,dx=\left\{{\begin{matrix}{\frac {\cos \left(\nu \arcsin {\frac {c}{a}}\right)}{\nu }}&a>c\\\\{\frac {a^{\nu }\,\cos {\frac {\nu \pi }{2}}}{\nu \,\left(c+{\sqrt {c^{2}-a^{2}}}\right)^{\nu }}}&a<c\end{matrix}}\right.\qquad {\text{Re}}(\nu )>0$

ohne Beweis