## 6.8 Functions Generated by Bessel Functions

### 6.81 Struve functions

6.811

1.

$\begin{array}{cc}{\int }_{0}^{\infty }{\text{H}}_{\nu }\left(bx\right)\text{d}x=-\frac{cot\left(\frac{\nu \pi }{2}\right)}{b}& \left[-20\right]\end{array}$

ET II 158(1)

2.

$\begin{array}{cc}\underset{0}{\overset{\infty }{\int }}{\text{H}}_{\nu }\left(\frac{{a}^{2}}{x}\right)\text{?}{\text{H}}_{\nu }\left(bx\right)\mathrm{d}x=-\frac{{J}_{2\nu }\left(2a\sqrt{b}\right)}{b}& \left[a>0,\text{?}\text{?}\text{?}\text{?}b>0,\text{?}\text{?}\text{?}Re\nu >-\frac{3}{2}\right]\end{array}$

ET II 170(37)

3.

$\begin{array}{cc}{\int }_{0}^{\infty }{\text{H}}_{\nu -1}\left(\frac{{a}^{2}}{x}\right)\text{?}{\text{H}}_{\nu }\left(bx\right)\frac{\mathrm{d}x}{x}=-\frac{1}{a\sqrt{b}}{J}_{2\nu -1}\left(2a\sqrt{b}\right)& \left[a>0,\text{?}\text{?}\text{?}b>0,\text{?}\text{?}\text{?}\text{?}\text{?}Re\nu >-\frac{1}{2}\right]\end{array}$

ET II 170(38)

6.812

1.

$\begin{array}{cc}{\int }_{0}^{\infty }\frac{{\text{H}}_{1}\left(bx\right)\text{d}x}{{x}^{2}+{a}^{2}}=\frac{\pi }{2a}\left[{I}_{1}\left(ab\right)-{L}_{1}\left(ab\right)\right]& \left[Rea>0,\text{?}\text{?}\text{?}\text{?}b>0\right]\end{array}$

ET II 158(6)

2.

$\underset{0}{\overset{\infty }{\int }}\frac{{\text{H}}_{\nu }\left(bx\right)}{{x}^{2}+{a}^{2}}\mathrm{d}x=-\frac{\pi }{2asin\left(\frac{\nu \pi }{2}\right)}{L}_{\nu }\left(ab\right)+\frac{bcot\left(\frac{\nu \pi }{2}\right)}{1-{\nu }^{2}}F_{1}2\left(1;\frac{3-\nu }{2};\frac{3+\nu }{2};\frac{{a}^{2}{b}^{2}}{2}\right)$

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