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Consistent with code
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paulmasson committed Nov 21, 2024
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<p>Successive branches differ by an overall complex factor, which can be motivated using the branch behavior of the logarithm:</p>

\[ [z^n]_k = [e^{n \log z}]_k = e^{n( \log z + 2k \pi i )} = e^{2k n \pi i} z^n \]
\[ [z^n]_k = [e^{n \log z}]_k = e^{n( \log z + 2k \pi i )} = e^{2nk \pi i} z^n \]

<p>An exponent with one decimal place is equivalent a fraction with ten in the denomiator, so that one expects it to take ten sheets to recapitulate the principal branch. The exceptions to this are integer exponents which have no denominator and hence one sheet, and exponents where the fraction reduces. For even numerators the exponent reduces to a multiple of one fifth and there are accordingly five distinct sheets. For half-integral exponents it reduces to a multiple one half and there are only two distinct sheets. If three decimal places were allowed as input it would in general take one hundred sheets to recapitulate the principal branch, unless the fraction again reduces.</p>

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