# Logarithmization of an integral

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• Aug 12th 2008, 12:25 AM
Bel Duarte
Logarithmization of an integral
Hi,

I wanna compute the integral of int_0^+inf exp(-1/2*(x+q)^2)*x^n *dx. First I logarithmized the expression which includes this integral. How can i handle its logarithmized?

Is it equal to int_0^+inf (-1/2*(x+q)^2+n*log(x))*1/x*dx? And the new limits of integration are they equal?

Any tip or reference is very welcome.

Bel Duarte
• Aug 12th 2008, 12:48 AM
wingless
No, you can't take the log of an integral, then integrate and exponantiate it again.

This integral doesn't have a solution in terms of elementary functions. But it can be written using Gamma function and a confluent hypergeometric function.
• Aug 12th 2008, 01:15 AM
Bel Duarte
Tx.

OK! There is no closed form for the integral. But i just want to logarithmize it and compute it, not to exponentiate it again. This integral is involved in the objective function of an optimization problem, and the same optimum is achieved taking the initial objective or its logarithm. Obviously the logarithmized form has several advantages.

Thanks again. Probably i have to compute it with a quadrature rule based on Laguerre polynomials.

Bel Duarte