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This question came up when I was reading through this question.

Are there definite integrals which cannot be computed using any real analysis techniques but are amenable using only complex analysis techniques?

If not, is there any reason to believe that if a definite integral can be evaluated using a complex analysis technique, then there must exist a way to compute the same definite integral using only real analysis techniques?

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Started a bounty for this question.

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abhIta
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    What is an example of a complex analysis technique that cannot be considered (a disguised instance of) a real analysis technique? – GEdgar Jun 26 '12 at 19:01
  • For instance, using Cauchy integral formula, method of residues, etc. – abhIta Jun 26 '12 at 19:05
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    So, now you need a rigorous definition of "real analysis method" that excludes these things. I don't know of one. – GEdgar Jun 26 '12 at 19:08
  • Interesting question! The challenge would be to define "real analysis technique" in a way that allows proof of existence of such a definite integral. – André Nicolas Jun 26 '12 at 19:55
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    Extract : "One time I boasted, "I can do by other methods any integral anybody else needs contour integration to do." So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow." Richard Feynman in "Surely You're Joking, Mr. Feynman!" – Raymond Manzoni Jun 26 '12 at 20:32
  • This does not seem like a well-posed question. – mrf Jun 29 '12 at 20:24
  • There are certainly real integrals which are much more easily computed with contour integration in the complex plane (for example, $\int_{-\infty}^\infty \frac{1}{1+x^n}$) but as the other comments suggest, I'm not sure that this methodology precludes being a technique of "real analysis." – rotskoff Jul 04 '12 at 16:49
  • An opposite question may be of interest http://math.stackexchange.com/questions/126655/when-a-real-valued-integral-cant-be-evaluated-using-contour-integration – Yrogirg Jul 06 '12 at 10:36

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Yes, there are so many such definite integrals that cannot be solved by real analysis techniques, such as

$$\int^{\infty}_{0}\frac{dx}{1+x^n}.$$

By the Cauchy's integral formula, we can compute it simply.

Vincenzo Tibullo
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mathon
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