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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 $\alpha$ ?

flag	asked Aug 28 at 7:25 pbs 153●5

fixed point theory should help you here. – S. Sra Aug 28 at 7:58

I would use Newton's Method. You can even let $a$ and $b$ be functions of $\alpha$. – Douglas Zare Aug 28 at 9:52

I'm looking for a closed-form solution. – pbs Aug 28 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 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 at 14:34

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Dox · Answer 1 · 2012-08-28 12:43:08Z UTC

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My first try would be to solve its differentiation. Then adjust possible $\alpha$ independent functions.

answered Aug 28 at 12:43

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

Techniques to solve equations involving a definite integral

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