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Is it possible to invert the following equation $$ e^{f(x)} = \int_{-\infty}^\infty dy_1 \dots \int_{-\infty}^\infty dy_{n-1} e^{g(x,y_1,\dots,y_{n-1})} $$

for $g(x,y_1,\dots,y_n)$, given some $f(x)$?

If analytic solutions are not possible, what types of numerical approaches are?

What type of integral equation does this fall under? It is somewhat similar to Fredholm integral equation of the first kind, but there is no identifiable kernel - even if I split $g$ into a kernel and an unknown function, solutions would yield the unknown function in terms of the kernel, rather than just $g$.

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  • $\begingroup$ It looks like a nonlinear integral operator, and so wouldn't fall under the Fredholm framework. Where did you find/come up with it? If it comes from a particular application domain, there may be work on it there. $\endgroup$ – Matt Mar 8 '17 at 19:19
  • $\begingroup$ Thanks for your comment. I came up with it from constructing approximations to the chemical master equation, so it's my fault in short. I had hoped the exponential form would yield some convenient special case $\endgroup$ – smörkex Mar 8 '17 at 20:26
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    $\begingroup$ Related: mathoverflow.net/questions/105689/… $\endgroup$ – smörkex Mar 8 '17 at 20:35

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