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Assume that $\zeta$ is a positive real number and $a = \frac{2 \pi}{\alpha_{\text{max}}}$ for $0 < \alpha_{\text{max}} < \frac{\pi}{2}$. In other words $a > 4$.

Is there a special function that when evaluated in a certain point is equal to

$$\int_0^{2 \pi} \textrm{e}^{i \zeta \cos(ax + \phi)} \, \sin^2(ax) \, \mathrm{d}x?$$

If $a$ would be a nice and an integer life would be good. Now I don't know!

  • 0
    Wouldn't it be easier to just say $a > 4$?2012-03-08
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    @RahulNarain Yes.2012-03-08
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    Downvoter... It is annoying.2012-03-08
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    The title "What special function is this?" does not tell much about the content of the post. Titles appear on the frontpage, Related sidebar, and on Google search results. The readers of such three places need a good, differential, idea on the content of the post.2012-03-09
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    I don't agree. If you don't know anything about special functions the addition will help nothing. Google search does not parse the LaTeX anyway.2012-03-09
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    I don't have sufficient time to look into this, but it seems to me that you'll need the [Anger-Weber functions](http://dlmf.nist.gov/11.10) for these. The $\phi$ term makes things a tad inconvenient...2012-03-19

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