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I've encountered the following beast in my research:

$$\frac{f(|x|)}{f(|x-\delta|)}=\operatorname{Exp}\left(a+bx^2-\frac{f(|x+\delta|)}{f(|x|)}\right)$$

Here, $x$ and $\delta\neq0$ are real numbers, and $a$ and $b$ are real constants.

I am wondering whether one can one write down $f(|x|)$ that satisfies the above in terms of functions of $|x|$, $a$, and $b$, removing $\delta$? I'll be happy with the solution involving special functions (hopefully ones that are implemented in Mathematica or MATLAB) -- I've tried using the Lambert W, but got stuck...

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up vote 1 down vote accepted

Your equation seems to have some problems. Assuming $b\neq 0$, if you set $\delta=0$ you get $\exp\left(a+bx^2\right)$ is a constant which contradicts $b\neq 0$.

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Oops. Good observation... I forgot the condition that $\delta\neq 0$... Will change the question. –  M.B.M. Sep 12 '12 at 5:37
    
I'm not sure if that would make much of a difference; at least for sufficiently regular functions. For instance, if $f$ is continuous and you tend $\delta$ to zero, you must have $e^{a+bx^2}=1\times e =e$ which cannot hold for all $x$ unless $b=0$ and $a=1$. –  S.B. Sep 12 '12 at 23:52
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