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Braches of Power Functions

The real part of successive branches of the power function \( z^n = ( x + i y )^n \) with noninteger exponent:

Each branch is rendered in two parts to keep the branch cut along the negative real axis clean.

Successive branches differ by an overall complex factor, which can be motivated using the branch behavior of the logaritm:

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

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.

This example is useful for thinking about the branches of the Bessel functions \( J_k(z) \) and \( I_k(z) \), both of which have power-function behavior about the origin.

This example requires the higher-level mathematics of Math.

Complete code for this example:

Examples Page