General condition of function $f$ satisfying $\int_{-\infty}^{\infty}e^{f(x,z)}dz = e^{g(x)}$

by Mikako   Last Updated October 20, 2019 05:20 AM - source

Suppose $x, z$ are real number, and suppose $f, g$ are real-valued function satisfying following equation

$\int_{-\infty}^{\infty}e^{f(x,z)}dz = e^{g(x)}$.

Could anyone find general condition of $f$ ?

I noticed $f(x,z) = \frac{1}{\sqrt{\pi}}e^{(x-z)^2}e^{g(x)}$ is an example of $f$ by using Gaussian integration, but I want to know general condition.

I asked similar question before, but my description was vague. Now I believe the description is enough to calculate general condition. If not, please comment. Thank you in advance!

Tags : integration

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