# 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? May 22, 2015 at 13:55
• @JackD'Aurizio Yes it is. It is order one half, correct! May 22, 2015 at 14:00
• How do you define $J_{\frac{1}{2}}(z)$ when $z<0$? May 22, 2015 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. May 22, 2015 at 15:49