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Do you know of any instance when a continued fraction is of help or necessary do describe a natural phenomenon?

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    $\begingroup$ This post (v2) seems like a list question. $\endgroup$ – Qmechanic Apr 8 '17 at 6:18
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    $\begingroup$ You may interest in the example here $\endgroup$ – Ng Chung Tak Apr 8 '17 at 9:54
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    $\begingroup$ One more use: method of continued fractions in quantum scattering theory. $\endgroup$ – Ruslan Apr 8 '17 at 14:39
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Here's a connection that I suspect minimizes the number of intermediate steps between the physics and the continued fractions:

If you want to study angular momentum, then you want to study the representation theory of $SU(2)$. This will force you to study Clebsch-Gordan coefficients, Wigner symbols, Racah symbols, etc. Locating the zeros of such symbols often requires solving one or more Pell equations (i.e. Diophantine equations of the form $X^2-nY^2=1$). The solutions to $X^2-nY^2=1$ form a free abelian group of rank 1, so to find all solutions, it suffices to find a generator of that group. That solution can be found as a convergent of the continued fraction for $\sqrt{n}$.

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  • $\begingroup$ Can you provide references for further reading on the topic? $\endgroup$ – Emilio Pisanty Apr 8 '17 at 11:12
  • $\begingroup$ @EmilioPisanty: Were you asking for references on the application of continued fractions to Pell's equation, or the application of Pell's equation to Wigner symbols, etc? $\endgroup$ – WillO Apr 8 '17 at 14:58
  • $\begingroup$ both topics, really. $\endgroup$ – Emilio Pisanty Apr 8 '17 at 15:44
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    $\begingroup$ @EmilioPisanty: The application of continued fractions to Pell's equation is standard fare in textbooks on number theory, and I'm sure there are a jillion to choose from. (I think I recall that Stark's book spends more time on this than most.) For the application of Pell's equation to representation theory, here is a pretty much randomly chosen example: iopscience.iop.org/article/10.1088/0305-4470/26/11/011 $\endgroup$ – WillO Apr 8 '17 at 16:09