Hello,


I am trying to solve an equation of the form


$C\_1 f(k\_1 z) + C\_2 f(k\_2 z) + C\_3 f(k\_3 z) + C\_4 f(k\_4 z) = C\_5 z^2$


for $f(z)$. Everything is complex. The $C\_i$'s and $k\_i$'s depend on some other parameters $\{ \lambda\_i \}$ through a very complicated dependence.


Any ideas?


Thanks!



How to solve a linear algebraic complex equation in one function evaluated at different arguments?
Make the change of the independent variable setting $g(z)=f(e^z)$ then $g(z+a)=f(ke^z)$ where
$k=e^a$. Now you have a linear difference equation. Difference equations have been well studied.
A general method is some kind of Fourier transform, look for a solution in the form
of exponential, then take a sum of those.
See, for example, [On equation f(z+1)-f(z)=f'(z)](https://mathoverflow.net/questions/114875), or the book of Gelfond, Calculus
of finite differences.


In your case, set $f(z)=az^2$. This will satisfy the equation if $\sum C\_jk\_j^2\neq 0$.


