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My math physics textbook requires me to prove an equation which is same as the linked two equations: https://dlmf.nist.gov/10.32#E8

I can prove similar integral representation of $I_\nu(x)$ (10.32.2 in the same link) using beta function, but this integral involves hyperbolic functions, so I'm lost.

The solution says that it is treated in the next section, and it really is, And the proof is just that confirming

(1) the expression satisfies the modified Bessel eq.,

(2) it has the small-z behavior required for $K_\nu$,

(3) and that it thas the required exponntially dacaying asymptotic value.

So it is not satisfying answer in two points: first that it is not elegant (it is not direct derivation), and second something feels wrong that a problem shows hyperbolic form first and no-hyperbolic form comes next, (and it was the case when I solve the problem for $I_\nu(x)$), but solution requires me to prove second eq. first.

Is it impossible to prove it directly, like that manipulating the given expression so it becomes the known expression of $K_\nu(x)$?

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