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Reproducing the Integral Definition of the Modified Bessel function


I need to simplify some integral expressions in terms of special functions, such as the modified Bessel function of the first kind. See for example Eq. (5) on this page. Notice that the real expressions I am working with are somewhat more complex, but I thought it might be a good idea to start by reproducing some known expression with Mathematica. However, I am failing miserably. After searching on the internal help pages and the web, I thought that this expression may do exactly what I wanted:


Assuming[{m \[Element] Integers && z \[Element] Reals}, 

 Integrate[Exp[z*Cos[x] ]*Cos[ m * x], {x, 0, Pi}]]

But it simply outputs the same integral again. What am I doing wrong? Why does it not output the result in terms of the Bessel function?




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