MHB Series representation for this integral

ConfusedCat
Messages
3
Reaction score
0
I am trying to find a series representation for the following expression
$$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}(b+x)^{-n}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ ; b,m,n,p are constant.

Is there a name for this function?

I found a series representation for $$\int_{i=0}^\infty {x^{\frac{2n-1}{2}}}e^{\left(-{\frac{x^2}{2m}}+\frac{x}{p}\right)} dx$$ in Table of Integrals, Series and Products by Gradshteyn and Ryzhik, involving parabolic cylinder function, but nothing that fits the first expression.

Any help would be appreciated.
 
Physics news on Phys.org
I would try writing power series representations for the exponential and (b+ x)^{-n}, multiply the series, then integrate term by term.
 
For original Zeta function, ζ(s)=1+1/2^s+1/3^s+1/4^s+... =1+e^(-slog2)+e^(-slog3)+e^(-slog4)+... , Re(s)>1 Riemann extended the Zeta function to the region where s≠1 using analytical extension. New Zeta function is in the form of contour integration, which appears simple but is actually more inconvenient to analyze than the original Zeta function. The original Zeta function already contains all the information about the distribution of prime numbers. So we only handle with original Zeta...

Similar threads

Back
Top