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Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\mathbb{C}$ is the unknown variable, and the expression is not an identity. Put another way, given the above expression are there techniques available to find the values of $\alpha$ for which the expression holds true, assuming we know from empirical study that there do exist such $\alpha$ ?

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Can this be any easier than finding all solutions to $p(x)=q(x)$ for known functions $p,q$? And can that be easier than finding solutions to $h(x)=0$ for known function $h$? – Gerry Myerson Aug 26 '12 at 10:38

It's a question slightly strange, but under certain not-too-tight conditions, we have $$\frac{d}{d\alpha}\left(\int_a^bf(x,\alpha)dx\right)=\int_a^b\frac{d}{d\alpha}\left(f(x,\alpha)\right)dx$$ So if you know both functions you could check whether $\,g'(\alpha)\,$ equals the above...

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It seems to be you are trying to find the arguments where two functions agree by finding the arguments where their derivatives agree. – Gerry Myerson Aug 26 '12 at 10:36
Well, there'll be equality up to an added constant...The question looks to me too general, but perhaps I'm missing something. – DonAntonio Aug 26 '12 at 10:46
We assume the expression is not an identity so it is not true for all values of $\alpha$. However, suppose it is true for some values of $\alpha$. I'm trying to find which values it is true for (in closed form, not just numerical points). So, I wondered if there were any standard techniques for doing this. – poirot Aug 26 '12 at 12:10

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