Evaluation of $L^2$ norm of exponential function on unit disk $\|\exp{z}\|_{L^2}$.

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While solving it using the definition of norm I got $\exp{2x}$ where $x=\mathfrak{R}(z)$. If I changed the integration to polar coordinate I was not able to do further.

We are being asked to find the solution in terms of Bessel's function which includes $J_1(2i)$. Reverse solving it using it's integral representation I was getting $0$ value.

Give some hints to proceed.

Thank you.