Can this integral be solved analytically?

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Discussion Overview

The discussion revolves around the possibility of solving a specific integral analytically. Participants explore various methods and resources for evaluating the integral, which involves exponential functions and a Gaussian term.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant expresses interest in finding an analytical solution to the integral, currently solved numerically.
  • Another suggests using Wolfram Alpha for assistance, although the initial poster reports issues with computation time on the website.
  • A participant proposes considering contour integrals and the residue theorem as potential methods for evaluation.
  • Further suggestions include trying Wolfram Alpha for an indefinite integral and consulting standard texts on integration techniques, such as Boros' "Irresistible Integrals" and Zwillinger's "The Handbook of Integration."
  • There is a mention of returning to numerical integration if analytical methods do not yield results.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the integral analytically, and multiple methods are proposed without agreement on their effectiveness.

Contextual Notes

Limitations include the potential complexity of the integral and the reliance on external computational tools, which may not provide satisfactory results.

Who May Find This Useful

This discussion may be useful for individuals interested in integral calculus, particularly those exploring analytical methods for complex integrals or seeking resources for integration techniques.

madness
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Hi everyone.

I have an integral I would like to solve. So far I've just been solving it numerically, but I wondered if it might be possible to get an analytical result.

Here it is:

<br /> \int_{-\infty}^{\infty} e^{k\cos(\omega t-a)}e^{-(bt-c)^2/(2\sigma^2)} dt<br />
 
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Try Wolfram Alpha - it will tell you!
 
Thanks for the tip, but I'm just getting "standard computation time exceeded" on the website version :). I wonder if there's any point looking at contour integrals and the residue theorem?
 
I would try Wolfram Alpha as an indefinite integral ... that should get a quick answer!

I think that would be worth a try - otherwise try one of the standard texts on integration techniques:
Boros' "Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals"
Zwillinger's "The Handbook of Integration"

Or that old standby, "Table of Integrals, Series and Products" by Gradshteyn and Ryzhik!

Else it is back to numerical integration.
 

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