Difficult integration: e^u(x^2)

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SUMMARY

The integral of the function g(x) = e^u(x) where u(x) = -(1-x^2)^(-1) from -1 to 1 presents significant challenges. The symmetry of the function allows for integration from 0 to 1, yet the user struggles with finding a suitable method for evaluation. Attempts to apply substitution and the Cauchy Integral Formula were unsuccessful, and even Mathematica resorts to Meijer G functions, indicating the complexity of the integral. The discussion highlights the need for advanced techniques in complex analysis and special functions to tackle such integrals.

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  • Understanding of integral calculus, particularly definite integrals.
  • Familiarity with complex analysis and the Cauchy Integral Formula.
  • Knowledge of special functions, including Meijer G functions.
  • Proficiency in using mathematical software like Mathematica for symbolic computation.
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  • Study the properties and applications of Meijer G functions in integration.
  • Learn advanced techniques in complex analysis, focusing on contour integration.
  • Explore substitution methods in integral calculus, particularly for non-elementary integrals.
  • Investigate the use of Dirac delta functions in relation to integrals of exponential functions.
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Mathematics students, researchers in applied mathematics, and anyone dealing with complex integrals and special functions will benefit from this discussion.

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Homework Statement



Given the function g(x)=e^u(x) where u(x) = -(1-x^2)^(-1). I have to integrate this from -1 to 1.

The Attempt at a Solution



I know the function is symmetric. It is enough to integrate it from 0 to 1 to get the real value of the integral. Well, beside that I have absolutely no clue how to do that. I need this in order to construct out of it a Dirac function. But my first task, as the homework states, is to solve this integral. (I tried to substitute something (but failed) and after that I wanted to use the Cauchy Integral Formule (extend the function to complex plane), but this didn't work either (because I couldn't get it in a appropriate form, as for the CIF needed)).

So I would be very pleased if someone can give me a hint. Perhaps a little bit more than a hint.
 
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I don't think there is a way to do it. Neither does Mathematica. (Well, it gives an answer in terms of Meijer G functions, but that seems excessive.)
 
maybe somehow incorporate the chain rule?
 

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