Can Bessel Functions Solve Specific Exponential Trigonometric Integrals?

[vvthuy]

Hi,

Do you have any idea to solve this integral?

\int^{\phi_{1}}_{\phi_{2}} exp[j cos(x)] dx

where \phi_{1} and \phi_{2} are an arbitrary angles. If \phi_{1}=\pi and \phi_{2}=0, the answer for this integral is a Bessel function.

Thanks,

Viet.

 

[Mute]

Hi,

Do you have any idea to solve this integral?

[tex]\int^{\phi_{1}}_{\phi_{2}} exp[j cos(x)] dx[/tex]

where \phi_{1} and \phi_{2} are an arbitrary angles. If \phi_{1}=\pi and \phi_{2}=0, the answer for this integral is a Bessel function.

Thanks,

Viet.

Use the identity

[tex]e^{ iz\cos\phi} = \sum_{s=-\infty}^{\infty}i^sJ_s(z)e^{is\phi}[/tex]

 

[vvthuy]

Use the identity

[tex]e^{ iz\cos\phi} = \sum_{s=-\infty}^{\infty}i^sJ_s(z)e^{is\phi}[/tex]

Thank you very much. Do you have the reference for this equation since I do not find it in my mathematical books?

 

[Mute]

I think the identity is listed in Abramowitz and Stegun:

http://www.math.sfu.ca/~cbm/aands/

(the website is the scanned pages of the reference book).