Proving an integration with a modified Bessel function and an exponential

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I am trying to prove the following identity:

enter image description here

where $\mu, h, H$, and $\tilde{\gamma}$ are real constants. The only hint that I have is use the relation between the modified bessel function of the first kind and the bessel function of the first kind.

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There is a typo in your equation, as shown below :

enter image description here