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I'm looking to integrate several functions having the form

$\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$

where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery of fractional calculus (e.g. fractional integration by parts) can be used to integrate such an expression.


(PS: I'm not quite sure to which category/tag this question belongs.)

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Is it a joke? The antiderivative of $\sin(a\tau)\tau^m$ is elementary for $m$ nonnegative integers only. Otherwise, integrate the corresponding power series. This isn't appropriate for MO. –  Wadim Zudilin Oct 15 '10 at 9:20
No it is not a joke. The problem seemed much harder to me at first and still does a bit. Specifically the possibility of antiderivatives of non-integer powers is still very new to me. Nevertheless, thanks for alerting me to both possibilities. –  Olumide Oct 15 '10 at 9:51
You are welcome, Olumide. I might be too strong in my opinion but it's not only the author who judges how hard is a problem. You can use integration by parts (which you can count as "fractional calculus") to reduce your problem to the case $0<m<1$. That's the elementary end of the story. –  Wadim Zudilin Oct 15 '10 at 10:23
Though not relevant here, Carlson's theorem (en.wikipedia.org/wiki/Carlson's_theorem) can sometimes be used to extend a formula for $f(n)$, where $n$ is an integer, to all complex numbers. A good example is the evaluation of Selberg's integral (e.g., page 2 of ms.unimelb.edu.au/~warnaar/pubs/Selberg_draft.pdf). –  Richard Stanley Oct 15 '10 at 17:19
@Wadim would you kindly name the technique that I should look at. I haven't studied fractional calculus yet but I'm willing to. However I'd like to solve integrate this expression as soon as possible in order to continue writing my thesis. Thanks. –  Olumide Oct 15 '10 at 23:06

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