Integral involving power , exp and exp(power)

[yfatehi]

I need some help evaluating the following integral from 0 to inifinity and a,p,c are positive reals, p>=1

\int\limit_0^\infty\ x^{p-1}e^{-x}e^{-c x^a} dx

 

[mathman]

In general it can be done only numerically - probably using a series expansion of e^-cxa.

 

[yfatehi]

How can use the series expansion

 

[mathman]

After the expansion each term will be e^-x times a power of x. If the powers are integers, then explicit term by term integration is possible. If they are not integers, I can't see anything but brute force numerical integration

 

[yfatehi]

If I have a power series is the variable x which is uniformaly absolutely convergent over the entire positive Reals domain and then I multipled this series by F(x) which does not depend on the series index n. ie all the series terms are multipiled by the same fuction. Will the resulting series preserve the convergence properties?

 

[mathman]

It will as long as |F(x)G(x)| is integrable, where G(x) is the function represented by the power series.

 

[yfatehi]

Plaese I need more information about this condition can you recommend me any reading material

 

[mathman]

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

See if this helps.