
Richard Feynman's Integral Trick - jorgenveisdal
https://medium.com/cantors-paradise/richard-feynmans-integral-trick-e7afae85e25c
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jiggawatts
Interestingly, 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
Interesting. Did you try MatLab?

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pmiller2
AKA Leibniz’s rule:
[https://en.wikipedia.org/wiki/Leibniz_integral_rule](https://en.wikipedia.org/wiki/Leibniz_integral_rule)

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olliej
Is there a link that doesn’t require signing in with Facebook, etc?

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hnakamura
Use a private window

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olliej
Thanks!

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crb002
If I recall he used power of two trapazoids to bound above and below?
Paywalled article.

~~~
laronian
Medium -> incognito

