4 ms·
Richard Feynman's Integral Trick
- pmiller2 7y agoAKA Leibniz’s rule: https://en.wikipedia.org/wiki/Leibniz_integral_rule https://en.wikipedia.org/wiki/Leibniz_integral_rule
- crb002 7y agoIf I recall he used power of two trapazoids to bound above and below? Paywalled article.
- laronian 7y agoMedium -> incognito
- jiggawatts 7y agoInterestingly, Mathematica does evaluate the integral successfully, but unless you add the constraint that alpha is real, it outputs a truly hideous looking result. This expression: Integrate[Log[1 - 2 \[Alpha] Cos[x] + \[Alpha]^2], {x, 0, \[Pi]}, Assumptions -> Abs[\[Alpha]] >= 1 && \[Alpha] \[Element] Reals] Required 33 seconds of CPU time @ 4 GHz and outputs the short and neat expression in the article. Slow, but successful. This is a symptom of Mathematica's symbolic engine being written with the "everything is a complex number" assumption deeply ingrained in it. Simplifying assumptions are treated as an afterthought. Conversely, it's not able to generalise to number spaces such as quaternions, matrices, or noncommutative algebras in general. It's a pity the Mathematica engine has not had a revamp, it would be great if they did a major update to bring its capabilities in line with modern group theory and category theory.
- jorgenveisdal 7y agoInteresting. Did you try MatLab?
- olliej 7y agoIs there a link that doesn’t require signing in with Facebook, etc?