Table of Integrals, Series, and Products, 8th Edition

6.8 Functions Generated by Bessel Functions

6.81 Struve functions

6.811

1. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }{\text{H}}_{\nu }(bx)\text{d}x=}-\frac{cot\left(\frac{\nu \pi }{2}\right)}{b}& \left[-2<Re\nu <0,\text{?}\text{?}\text{?}\text{?}b>0\right]\end{array}$

  ET II 158(1)

2. 

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

  ET II 170(37)

3. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }{\text{H}}_{\nu -1}\left(\frac{a^2}{x}\right)\text{?}{\text{H}}_{\nu }(bx)\frac{\mathrm{d}x}{x}=-\frac{1}{a\sqrt{b}}{J}_{2\nu -1}\left(2a\sqrt{b}\right)}& \left[a>0,\text{?}\text{?}\text{?}b>0,\text{?}\text{?}\text{?}\text{?}\text{?}Re\nu >-\frac{1}{2}\right]\end{array}$

  ET II 170(38)

6.812

1. 

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }\frac{{\text{H}}_1(bx)\text{d}x}{x^2+a^2}=\frac{\pi }{2a}\left[I_1(ab)-L_1(ab)\right]}& \left[Rea>0,\text{?}\text{?}\text{?}\text{?}b>0\right]\end{array}$

  ET II 158(6)

2. 

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