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When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means convolution as in $(f*g)(x)=\int\limits_{\mathbb R}f(y)g(x-y)\,dy$)

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    even when $g=0$ the equation does not always admits solutions analytic near 0 (take $h=Id$). it seems pretty complicated...2011-11-29
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    Sounds like an integral equation to me...2011-11-29
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    @J.M.: in the most general sense yes, though I think that label usually means linearity in $f$. So, is this some kind of "no"?2011-11-29
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    Not really. There is such a thing as a nonlinear integral equation after all...2011-11-29
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    @Glougloubarbaki: I'm fine with multi-valued, non-entire solutions, but I guess you're right about it being complicated. Maybe only special choices of $g,h$ permit analytic expressions...2011-11-29
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    @J.M.: True. So, are there any chances on an exact solution or do I have to stick to iteration/numerics?2011-11-29
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    Unfortunately I've no experience with nonlinears. Sorry... :(2011-11-29
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    If you mean convolution, you should use $(f*g)(x)$ (or $f*g(x)$) instead of $f(x)*g(x)$ which makes (strictly spoken) no other sense then multiplication.2011-11-29
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    @Dirk: thanks, I fixed it2011-11-29
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    @J.M. no problem, thanks for reminding me that I should search for integral equations2011-11-29
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    You should probably say what you mean by *convolution* as well. For example, $\int_{-\infty}^\infty f(x-t)g(t)\,dt$ or $\sum_{k=0}^x f(x-k)g(k)$ or what?2011-11-29
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    @GEdgar You're right, I'll specify that I mean the $\int_{-\infty}^\infty$ one2011-11-30
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    So the variable $x$ is real? Are the values real? When you say "analytically" do you mean it should be an analytic function? Or is that a vague way of saying "in closed form"?2011-11-30
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    @GEdgar: $x$ is real, $f,g,h$ are complex valued (in fact, at least $f$ is analytic), and "yes", with analytically I mean in closed form (so I'll reword that, thanks for pointing it out)2011-11-30
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    This is a very interesting question? Have you considered posting bounty on it?2015-01-20

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