Evaluate a Trigonometric Integral

[jackychenp]

Hi All,

\int_{0}^{2\pi}d\varphi \int_{0}^{\pi}\sqrt{\frac{1}{1-sin^2\theta cos^2\varphi }}d\theta
I tried very hard to evaluate this integral but failed. Please help me if you know how to solve it.

 

[JJacquelin]

This integral cannot be expressed in terms of a finite number of elementary functions.
Analytic expression requires a kind of special functions called "Elliptic Integrals". More precisely In the present case : K(x) the complete elliptic integral of the first kind. Of course, the result can also be expressed on the form of infinite series. See :
http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html
http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html

 

[jackychenp]

Thanks a lot!

This integral cannot be expressed in terms of a finite number of elementary functions.
Analytic expression requires a kind of special functions called "Elliptic Integrals". More precisely In the present case : K(x) the complete elliptic integral of the first kind. Of course, the result can also be expressed on the form of infinite series. See :
http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html
http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html