Physical Meaning and Evaluation of Complex Integral in Heat Conduction Problems

[JulieK]

Hello
In the following integral which I obtained from Wolfram, what is the physical meaning (e.g. in a heat conduction problem) of $$\sqrt{-\sinh^{2}(bx)}$$. Also how to evaluate this at $$x=L$$.
Thanks.

$$\int\frac{dx}{\cosh^{3n+1}(bx)}=\frac{\sinh(bx)\cosh^{-3n}(bx)}{3nb\sqrt{-\sinh^{2}(bx)}}\,_{2}F_{1}\left(\frac{1}{2},-\frac{3n}{2};\frac{2-3n}{2};\cosh^{2}(bx)\right)$$

 

[Gib Z]

Welcome to PF!

The antiderivative of a real function can not possibly be complex, So perhaps the hypergeometric function has imaginary terms that cancel it out.

 

[jackmell]

That is indeed the case as the parameters to the Hypergeometric function force it into it's analytic continuation giving rise to a imaginary number which cancels the i in the denominator.