Timeline for Improper integral $\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$
Current License: CC BY-SA 3.0
12 events
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| May 22, 2014 at 2:14 | comment | added | Tom Dickens | @user113103: I have run across cases where I could get Mathematica to compute the needed Mellin transforms (sometimes by experimenting with changes of integration variable) and then could work out the integral from there. So it IS possible! | |
| May 21, 2014 at 15:55 | comment | added | user113103 | @TomDickens: Thanks for the suggestion. I will look into the paper, and give it a try. If mathematica uses this method internally but can't evaluate the integral, do I have a chance to get to a closed form by doing it by myself? | |
| May 21, 2014 at 15:53 | vote | accept | user113103 | ||
| May 20, 2014 at 21:03 | comment | added | Lucian | @TomDickens: Correct! Thanks for pointing that out! | |
| May 20, 2014 at 20:40 | answer | added | Bazin | timeline score: 4 | |
| May 20, 2014 at 20:08 | comment | added | Tom Dickens | By the way, Liouville's theorem and the Risch algorithm are about closed-form antiderivatives, not definite integrals. | |
| May 20, 2014 at 20:07 | comment | added | Tom Dickens | I would try using a Mellin transform approach. You can evaluate the Mellin transform of both functions in your integrand (I was able to using Mathematica, taking 1+x^2 -> u^2) in cosh term), then use the convolution theorem to write the integral over a vertical line in the complex plane. The trick then is to close the contour and evaluate the integral as the sum of residues from the poles of Gamma functions. There are many references to this, a good paper is "Evaluation of integrals and the mellin transform," by Prudnikov, et al. Mathematica uses this technique internally. | |
| May 20, 2014 at 18:41 | answer | added | Lucian | timeline score: 3 | |
| May 20, 2014 at 15:11 | comment | added | Neil Strickland | In the case $a=b=1$ I calculated the integral to 50 digits using Maple and gave the result to the Inverse Symbolic Calculator at oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html; it was not recognized. Given that neither Maple, Mathematica nor the ISC knows an answer in closed form, I doubt that one exists. | |
| May 20, 2014 at 14:31 | review | First posts | |||
| May 20, 2014 at 15:15 | |||||
| May 20, 2014 at 14:30 | history | edited | user113103 | CC BY-SA 3.0 |
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| May 20, 2014 at 14:13 | history | asked | user113103 | CC BY-SA 3.0 |