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I found the following integral in a paper I was reading: \begin{equation} \frac{1}{2\pi} \int\limits_{-\pi}^{\pi} \exp\left(a \cos x + b \sin x\right) dx = I_0\left(\sqrt{a^2+b^2}\right), \end{equation} where $I_0$ is the modified Bessel function of the first kind. Unfortunately, there was no reference. I tried to verify the integral with Mathematica, but with no result. I also spend some time to find it in Gradshteyn/Ryzhik, again with no result. Is the above integral correct (including some reference or justification)? Thanks in advance.

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The definition of the modified Bessel function $I_0$ is $$ I_0(z)=\frac{1}{\pi}\int_0^\pi e^{z\cos x}dx=\frac{1}{2\,\pi}\int_{-\pi}^\pi e^{z\cos x}dx $$ Now $$ a\cos x+b\sin x=\sqrt{a^2+b^2}\cos(x+\phi) $$ for some angle $\phi$. Since the integral is over a full period, you get the result.

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  • $\begingroup$ That turned out to be surprisingly simple. Thank you! $\endgroup$ – TriSSSe Mar 2 '13 at 13:35

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