# Integrals with bessel functions

• areslagae

#### areslagae

I am trying to solve

int(int(exp(a*cos(theta)*sin(phi))*sin(phi), phi = 0 .. Pi), theta = 0 .. 2*Pi) (1)

with a a constant.

Using the second last definite integral on

http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions

the integral (1) reduces to

2*Pi*(int(sin(phi)*BesselI(0, a*sin(phi)), phi = 0 .. Pi)) (2)

Can anyone solve (1) or (2)?

I am trying to solve

int(int(exp(a*cos(theta)*sin(phi))*sin(phi), phi = 0 .. Pi), theta = 0 .. 2*Pi) (1)

with a a constant.

Using the second last definite integral on

http://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions

the integral (1) reduces to

2*Pi*(int(sin(phi)*BesselI(0, a*sin(phi)), phi = 0 .. Pi)) (2)

Can anyone solve (1) or (2)?

Plugging it into Mathematica assuming a>0 gives 4*Pi*Sinh[a]/a.

It doesn't tell the steps used, unfortunately :)

Thanks!

Apparently, Mathematica is the only program that can solve several of the integrals I am dealing without of the box. Unfortunately, I do not have access to Mathematica.

Would you please be so kind to try if Mathematica can solve

int(exp(a*cos(phi))*(sin(phi))^2, phi = 0 .. Pi)

with a a positive real constant?

The integral

assume(a > 0);
int(exp(a*cos(phi))*sin(phi)^2, phi = 0 .. Pi);

equals to

(Pi/a)*BesselI(1,a)

I have solved the integral using Mathematica, which seems to solve all these integrals out of the box.