More exactly, can someone help me solve this?

http://rogercortesi.com/eqn/tempimagedir/eqn4066.png

Hi Tedor,

I'm not a mathematician. But I use wolfram alpha to solve many math problems, So I gave it a try. Here is what I got for indefinite integral. (It has no standard method but can be expanded as a series) . When limits are applied it timed out sadly. But you may be able to use it to some extend

Here are the links for solutions.

https://www.wolframalpha.com/input/?i=intergrate%28%28e^%28-x%28x-mu%29^2%29%2Fsqrt%28alpha^2-x^2%29%29

https://www.wolframalpha.com/input/?i=intergrate%28%28e^%28-x%28x-mu%29^2%29%2Fsqrt%28alpha^2-x^2%29%29%29+from+-infinity+to+%2Binfinity

And I add the results as images

Gayan

If you use the following change of variable

t = x / alpha

then the interval of integration becomes [-1, 1] and the integrand will take the more convenient form

G(t) / sqrt(1 - t²)

which is adequate for an approximation via a N-points Gauss-Chebyshev quadrature..

The result is A*Sum ( G(t_k) ) with k = 1...N

where A = pi / N

and t_k = cos( A(k - 0.5) )

The integrand is an even function only in the special case where the parameter u = 0.

@H.E. you are right, I think you mean when mu=0.

Thanks Demetris,

Yes indeed, I have used "u" instead of "mu". That's because I was answering your post, which, for some reason, does not appear anymore and in which you have used "a" instead of "alpha" (LOL).

More seriously, there is a real need for some editing tools for symbols and equations.

Cheers.

I've met this quesrtion/answers occationally. The proposed solution is not fair: a) There is omitted dependence of I(mu, alfa) on mu. If mu is not equal to 0, I(...) depends on both parameters; b) Two poles in x =.+ - alfa lay inside of the intregration interval, and correct solution requires the residuals technique to be used. The best solution is to search the integral in Gradshteyn and Ryzhik's "Table of Integrals, Series, and Products".

A demonstration of Chebyshev–Gauss quadrature method for this integral and for the case alpha=2, mu=1 is the attachment file.

For 10 digits accuracy we need n=10 point approximation.

Good work H.E.!

Thank you everyone. It seems that the integral itself is transcendental, and I haven't found a way to solve it exactly. I have tried to determine the pdf and the cdf for a mixed signal, a sinus CW + gaussian noise. The integral comes from the convolution of the two pdf`s. A Taylor series approach woldn`t be helpful since I need an asimptotic solution rather than a local one.

I have attached a complete formulation of the problem.

Here is the complete description of the problem

Dear Teodor,

I followed the guides of the description and nothing good was created.

Instead of them, I took a Taylor polynomial approximation of N(0,sigma^2) and I kept the second distribution as it was.

The PDF of the sum of the two random variables was found and has enough accuracy.

See results at the attachment.

It is too approximate and almost desperate analysis, but it was the only that really worked.

Good luck!

Nevertheless, I would advise you to look at the Gradshteyn and Ryzhik's "Table of Integrals, Series, and Products". This is the greatest in the world and well systematized collection of integrals, including the most terrible, verified not by one generation of mathematicians and phisicists. It should be in the library.

Dear Anatoly, I looked the book and I found it great!

Thanks for suggesting it.

That's fine. This book is unique and really useful. Wish all participants of discussion further successes!