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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$\begingroup$ fixed point theory should help you here. $\endgroup$– SuvritCommented Aug 28, 2012 at 7:58
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$\begingroup$ I would use Newton's Method. You can even let $a$ and $b$ be functions of $\alpha$. $\endgroup$– Douglas ZareCommented Aug 28, 2012 at 9:52
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$\begingroup$ I'm looking for a closed-form solution. $\endgroup$– pbsCommented Aug 28, 2012 at 12:45
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3$\begingroup$ 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. $\endgroup$– Deane YangCommented Aug 28, 2012 at 13:42
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1$\begingroup$ 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 $\alpha$. $\endgroup$– Gerald EdgarCommented Aug 28, 2012 at 14:34
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The problem space of symbolic computation on definite integrals is currently fairly open. The Risch algorithm answers the question if there is a closed form solution to a indefinite integral (assuming it doesn't get stuck on the constant problem) in elementary functions, but it doesn't address solutions with special functions. So if there is an antiderivitive for f(x), Risch algorithm will find it and it will tell you if there isn't one.
If there isn't an antiderivitive expressible in elementary functions, we need to resort to heuristics to try to match patterns; We try to search for special functions that match, especially the incomplete gamma function. While we have rules of thumb to search for symbolic solutions to definite integrals, we can't always say for sure whether a symbolic solution exists or what it is.
This is an active area of research for developers of computer algebra systems, and these are hard problems. Even the Risch algorithm, which is fairly mature in academic terms, is difficult to understand and fully implement, and to date no CAS fully implements it.
Techniques for symbolic integration on definite integrals for humans often start with look up tables. For machines, you want to start with Risch, then go from there. There are pattern matching techniques for special functions, then there are techniques that employ some of the search space techniques of automated theorem provers.
The Wolfram article covers some of the reasons why definite integration is still so difficult; Concise techniques for providing closed form solutions or answers to if there is no closed form solutions would solve a number of open problems in transcendence theory. If we find an algorithmic technique for definite integrals, even if its only semi-decidable as Risch is, it will be a major step in a number of areas of mathematics.
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1$\begingroup$ According to another MO answer, Axiom's implementation does not fully implement the Risch algorithm. $\endgroup$ Commented Dec 3, 2020 at 17:27
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$\begingroup$ @user541686, before attempting to perform your edit, have you also read the comments under the answer that you are linking to? I bet you haven't. $\endgroup$– Alex M.Commented Mar 30, 2021 at 10:03
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$\begingroup$ @user541686: Those comments say two things: 1) that CASs become better with every version (with a concrete example given therein); 2) that what some CAS cannot compute at a given moment in time, some other one could. From these two (and that post) it is impossible to deduce that "no CAS fully implements the Risch algorithm". $\endgroup$– Alex M.Commented Mar 30, 2021 at 12:53
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$\begingroup$ @AlexM. Actually, "no CAS fully implements the Risch algorithm" is what the linked answer is asserting. The Risch algorithm does provide a complete decision process for mixed algebraic and transcendental functions (provided the underlying field is something like the rational numbers, which has decidable zero-recognition). $\endgroup$ Commented Apr 1 at 13:03