Riemann Surfaces

Examples

The following are some examples of multiple-valued functions. In each case, the branch is identified with a different color. Click on either the top-right menu or the following functions to explore.

$f(z) = z^{1/2}$ $f(z) = z^{1/3}$ $f(z) = \sqrt{1-z^2}$ $f(z) = \dfrac{1}{\sqrt{1-z^2}}$ $f(z) = \arctan(z)$

Real component of $f(z)=z^{1/2}$

Imaginary component of $f(z)=z^{1/2}$

Real component of $f(z)=z^{1/3}$

Imaginary component of $f(z)=z^{1/3}$

Real component of $f(z)=\sqrt{1-z^2}$

Imaginary component of $f(z)=\sqrt{1-z^2}$

Real component of $f(z)=\frac{1}{\sqrt{1-z^2}}$

Imaginary component of $f(z)=\frac{1}{\sqrt{1-z^2}}$

Real component of $f(z)=\arctan(z)$

Imaginary component of $f(z)=\arctan(z)$

Riemann Surfaces

Examples

Real component of \(f(z)=z^{1/2}\)

Imaginary component of \(f(z)=z^{1/2}\)

Real component of $f(z)=z^{1/3}$

Imaginary component of $f(z)=z^{1/3}$

Real component of \(f(z)=\sqrt{1-z^2}\)

Imaginary component of \(f(z)=\sqrt{1-z^2}\)

Real component of \(f(z)=\frac{1}{\sqrt{1-z^2}}\)

Imaginary component of \(f(z)=\frac{1}{\sqrt{1-z^2}}\)

Real component of \(f(z)=\arctan(z)\)

Imaginary component of \(f(z)=\arctan(z)\)