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

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2.1

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

ohne Beweis (Weber)

 

2.2

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

ohne Beweis

 

3.1

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

ohne Beweis

 

3.2

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

ohne Beweis

 

3.3

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

ohne Beweis