Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $\alpha$ ?

share|improve this question
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
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
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
show 2 more comments

1 Answer

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

share|improve this answer
But just because the equation holds doesn't mean its derivative does. Or vice versa. –  Deane Yang Aug 28 '12 at 13:40
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.