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 $\alpha$ ?
General question about solving equations involving a definite integral
2
$\begingroup$
integral-equations
-
2Can 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 $h$? – 2012-08-26