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

Zurück zu Bestimmte Integrale

2.1

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

2.2

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

3.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{2n}(ax)\frac{\sin (cx)}{x}dx=\{\begin{array}{cc}0& a<c\\ \\ \frac{(-1{)}^{n-1}}{2n}{U}_{2n-1}\left(\frac{c}{a}\right)\sqrt{1-{\left(\frac{c}{a}\right)}^2}& a>c\end{array}$

3.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{2n+1}(ax)\frac{\sin (cx)}{x}dx=\{\begin{array}{cc}\frac{(-1{)}^n}{2n+1}(T_{2n+1}\left(\frac{c}{a}\right)-U_{2n}\left(\frac{c}{a}\right)\sqrt{{\left(\frac{c}{a}\right)}^2-1})& a<c\\ \\ \frac{(-1{)}^n}{2n+1}{T}_{2n+1}\left(\frac{c}{a}\right)& a>c\end{array}$

3.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{\nu }(ax)\frac{\sin (cx)}{x}dx=\{\begin{array}{cc}\frac{\sin (\nu \arcsin \frac{c}{a})}{\nu }& a>c\\ \\ \frac{a^{\nu }\sin \frac{\nu \pi }{2}}{\nu {(c+\sqrt{c^2-a^2})}^{\nu }}& a<c\end{array}\text{Re}(\nu )>-1$