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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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fixed point theory should help you here. –  Suvrit Aug 28 '12 at 7:58
    
I would use Newton's Method. You can even let $a$ and $b$ be functions of $\alpha$. –  Douglas Zare Aug 28 '12 at 9:52
    
I'm looking for a closed-form solution. –  pbs Aug 28 '12 at 12:45
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I'm not sure I understand. You want to find a closed-form solution, even though you can't find a closed-form formula for the integral? You might have better luck if you can post a specific example that you want to solve. –  Deane Yang Aug 28 '12 at 13:42
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So, a special case is $f(\alpha)=g(\alpha)$, solve for $\alpha$? And you expect a "closed form" solution? Say, Kepler's equation, $M = \alpha - \epsilon \sin \alpha$, solve for $\alpha$. –  Gerald Edgar Aug 28 '12 at 14:34
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My first try would be to solve its differentiation. Then adjust possible $\alpha$ independent functions.

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But just because the equation holds doesn't mean its derivative does. Or vice versa. –  Deane Yang Aug 28 '12 at 13:40
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