The real and imaginary parts of an inverse function on the complex plane.

For fractional values of the exponent only the principle sheet is displayed. Limits are set on the z-axis to trim the infinite peaks at poles.

The inverse function is

$f(z) = \frac{ 1 }{ 1 - z^n }$

With the parametrization $$z = r e^{ i \phi }$$ the function can be written

\begin{align} f(z) &= \frac{ 1 }{ 1 - r \cos n\phi + i r \sin n\phi } \\ &= \frac{ 1 - r \cos n\phi }{ ( 1 - r \cos n\phi )^2 + r^2 \sin^2 n\phi } - i \frac{ r \sin n\phi }{ ( 1 - r \cos n\phi )^2 + r^2 \sin^2 n\phi } \end{align}

so that evaluations can be done with standard JavaScript functions. More complicated functions would require a library that can handle the arithmetic of complex numbers, such as Math.

Complete code for this example:

Examples Page