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If I remember rightly there are some integrals of real functions which are easier to compute by using complex analysis.

Is this because of properties of the particular function or because of a lack of a known real analysis technique?

Are there functions which would require hypercomplex analysis to integrate?

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    This question is extremely vague2010-07-27
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    My understanding of the question is: "There are definite integrals over parts of the real line that are easier to compute using techniques from complex analysis. What properties of the function are responsible for this? Are there any integrals that cannot be computed without further extensions with hypercomplex numbers?" This sounds fine to me.2010-07-27
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    There is a famous anecdote of Feynman in which he recalls competing with other people to evaluate integrals. Feynman would use differentiation under the integral sign, and he claimed that this technique (which his colleagues never used) worked on any integral. But one of his colleagues showed him an integral he could not integrate; the only method his colleague knew that worked was complex analysis. Unfortunately I don't know the integral in question.2010-07-27

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