Complex Analysis

Riemann Surfaces

Examples


The following are some examples of multiple-valued functions. In each case, the branch is identified with a diferent color. Click on the following functions or scroll down 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 \(f(z)=z^{1/2}\)

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




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

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




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

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




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

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




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

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Note: All applets made with MathCell created by Paul Masson. The source code is available here.


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