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I know the Bessel function can be formatted as follows. $$J_n\left(x\right)=\frac{1}{\pi}\int_{0}^{\pi}\text{cos}\left(n\tau-x\text{sin}\tau\right)d\tau$$ So can we get a Bessel-similar expression for the integration in the title? Many thanks!

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  • $\begingroup$ Have you looked on Wikipedia? $\endgroup$ – WA Don May 21 '20 at 13:39
  • $\begingroup$ @WADon I have checked this Wiki page, but I think I didn't find direct answer to this question. I have also tried some manipulations, but all failed. I want to express this integration in the Bessel function form, so I tried different approaches to convert it to Bessel form. $\endgroup$ – Jason Zhou May 21 '20 at 13:52
  • $\begingroup$ You are right - there is a similar integral similar for $Y_n(t)$ but not quite what you needed. Sorry for sending you on a needless search. $\endgroup$ – WA Don May 21 '20 at 14:15
  • $\begingroup$ @WADon That's all right, thanks for spending your time! $\endgroup$ – Jason Zhou May 21 '20 at 14:22
  • $\begingroup$ I am pretty sure you will get a solution, which is a sum of Struve functions $H_\alpha(x)$, since for $n=0$ your integral leads to $-\pi\,H_0(x)$, assuming $x\in\mathbb{R}$. For $n=1$ we get $-\pi\,H_{-1}(x)$. For higher $n$ we get more and more involved results. I guess that using $\sin(n \tau - x \sin(\tau))=\cos(x\, \sin(\tau)) \sin(n \tau) - \cos(n \tau) \sin(x \sin(\tau))$ and the general expansion formulas for $\sin(n \tau)$ and $\cos(n \tau)$ can do the trick. You will end up with a sum of Struve functions. Sorry I can not be of more help. $\endgroup$ – Michael_K May 21 '20 at 19:57
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According to https://en.wikipedia.org/wiki/Anger_function, https://dlmf.nist.gov/11.10,

$\int_0^\pi\sin(n\tau-x\sin\tau)~d\tau=\pi\mathbf{E}_n(x)$

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