New to the method of steepest descent

[Beer-monster]

Homework Statement

I'm new to this approximation method and was wondering the best way to proceed with this function:

$$\int_{-\infty}^{+\infty} dx e^{\frac{ax^{2}}{2}}e^{ln[2cosh(b+cx)]}$$

I've found the saddle point (I think). But I was wondering if it would be best to expand the x squared term or the ln(cosh) term. If the latter, should I expand the cosh as a taylor series to get ln(expanded cosh) and then expand the logarithm (of the expanded cosh?) and simplify the result?.

Thanks for your help.

 

[murmillo]

Sorry--I'm not quite sure what the problem you're trying to answer is.

 

[Beer-monster]

I'm not quite sure how to integraye this function using the method of steepest descent. Usually you have a function of a complex variable, which this is not.

And often examples show only one exponential function where I have a product of two, so I'm not 100% which is the more rapidly varying one (i.e. which function do I expand as a Taylor series).

I went ahead and worked with the ln(cosh) function, and calculated its derivatives to get the Taylor series to 2nd order. Now I'm not sure how to move forward? Do I just integrate or do I need to substitute complex variables?