# Thread: integration of multiplication of Gaussian and Lorentzian

1. ## integration of multiplication of Gaussian and Lorentzian

I need to evaluate very generally the following integration,

$\int_{-\infty}^{\infty}\frac{1}{x^{2}+a^{2}}e^{-{\frac{(x-x_{0})^{2}}{2\sigma^{2}}}}\mathrm{d}x$

Is there analytic expression for that? Thanks,

2. Originally Posted by bsmile
I need to evaluate very generally the following integration,

$\int_{-\infty}^{\infty}\frac{1}{x^{2}+a^{2}}e^{-{\frac{(x-x_{0})^{2}}{2\sigma^{2}}}}\mathrm{d}x$

Is there analytic expression for that? Thanks,
Well WolframAlpha does not know what it is.

CB

3. Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!

4. Originally Posted by bsmile
Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!
Since there is an algorithm for doing integrals (Risch algorithm) that modern software does not return a closed for for an integral generally means that a closed for does not exist in terms of elementary functions.

CB

5. Originally Posted by bsmile
Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!
Quite right. What was the Captain thinking .....?

In fact a solution does exist: There is a special function called the Fantastic F-function of the Second Kind that's exactly the answer to your problem. The properties of this function are left for you to explore as I don't think much literature on it exists at the moment.

(The Fantastic F-function of the First Kind was suggested to someone quite a few years ago now as the answer to an unrelated problem. No doubt a function of the Third Kind will be suggested in the future to someone else).

6. Originally Posted by mr fantastic
Quite right. What was the Captain thinking .....?

In fact a solution does exist: There is a special function called the Fantastic F-function of the Second Kind that's exactly the answer to your problem. The properties of this function are left for you to explore as I don't think much literature on it exists at the moment.

(The Fantastic F-function of the First Kind was suggested to someone quite a few years ago now as the answer to an unrelated problem. No doubt a function of the Third Kind will be suggested in the future to someone else).
You are joking, right?! I am desperately looking for a closed form solution for that integral to save cpu time, and was glad to see the strange F-function, but a second thought realizes that it does not exist at the time being ....

Originally Posted by CaptainBlack
Since there is an algorithm for doing integrals (Risch algorithm) that modern software does not return a closed for for an integral generally means that a closed for does not exist in terms of elementary functions.

CB

Seems there are really very good mathematicians on this forum, , thanks for letting me know the Risch algorithm ...

7. Originally Posted by bsmile
You are joking, right?! I am desperately looking for a closed form solution for that integral to save cpu time, and was glad to see the strange F-function, but a second thought realizes that it does not exist at the time being ....
Necessity is the mother of invention (of non-elementary functions).