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

Zurück zu Bestimmte Integrale

1.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x}{\sqrt{{\alpha }^2+x^2}}{J}_0(x)dx=e^{-\alpha }\text{Re}(\alpha )\ge 0$

1.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\alpha }{\sqrt{{\alpha }^2+x^2}}{J}_1(x)dx=1-e^{-\alpha }\text{Re}(\alpha )\ge 0$

2.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{\nu }(x)x^{s-1}dx=\frac{2^{s-1}\mathrm{\Gamma }\left(\frac{\nu +s}{2}\right)}{\mathrm{\Gamma }(1+\frac{\nu -s}{2})}-\text{Re}(\nu )<\text{Re}(s)<\frac{3}{2}$

3.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{J}_{\mu }(x)J_{\nu }(x)x^{s-1}dx=2^{s-1}\frac{\mathrm{\Gamma }(1-s)\cdot \mathrm{\Gamma }\left(\frac{\mu +\nu +s}{2}\right)}{\mathrm{\Gamma }(1+\frac{\mu -\nu -s}{2})\cdot \mathrm{\Gamma }(1+\frac{\nu -\mu -s}{2})\cdot \mathrm{\Gamma }(1+\frac{\mu +\nu -s}{2})}$