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

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2.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_0(ax)\frac{1-\cos (cx)}{x^2}dx=\{\begin{array}{cc}\frac{c\pi }{2}-a& a<c\\ \\ c\cdot \arcsin \left(\frac{c}{a}\right)+\sqrt{a^2-c^2}-a& a>c\end{array}$

2.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_0(ax)\frac{1-\cos (cx)}{x}dx=\{\begin{array}{cc}\text{arcosh}\left(\frac{c}{a}\right)& a<c\\ \\ 0& a>c\end{array}$

2.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_0(ax)\cos (cx)dx=\{\begin{array}{cc}0& a<c\\ \\ \frac{1}{\sqrt{a^2-c^2}}& a>c\end{array}$

3.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{\nu }(ax)\frac{\cos (cx)}{x}dx=\{\begin{array}{cc}\frac{\cos (\nu \arcsin \frac{c}{a})}{\nu }& a>c\\ \\ \frac{a^{\nu }\cos \frac{\nu \pi }{2}}{\nu {(c+\sqrt{c^2-a^2})}^{\nu }}& a<c\end{array}\text{Re}(\nu )>0$