# Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral:

$$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,j_{0}(b\cos x)\,j_{0}(b\sin x)\,\mathrm dx$$

where $a,b$ are positive real constants and $j_{0}$ is the spherical bessel function of order zero ($j_0(z)=sinc(z)$). Anyone has a clever substituion or a known integral that might help me in finding the solution?

• Really strange. One might guess that $J_{0.5}(z)$ is the Bessel function of the first kind of order one half, also known as: $$J_{\frac{1}{2}}(z)=\sqrt{\frac{2}{\pi z}}\sin(z).$$ Do our notations agree? – Jack D'Aurizio May 22 '15 at 13:55
• @JackD'Aurizio Yes it is. It is order one half, correct! – JFNJr May 22 '15 at 14:00
• How do you define $J_{\frac{1}{2}}(z)$ when $z<0$? – Jack D'Aurizio May 22 '15 at 15:43
• @JackD'Aurizio yes you are right again...I really confused the notations for bessel function and spherical bessel function. The text is now edited in the right way. – JFNJr May 22 '15 at 15:49