Calculus Small Changes, Big Ideas

Section 16.3 Visualizing complex functions

• Domain coloring: Let the input be the point on the complex plane, and let the output be how we color it.

  • Lightness = modulus (black = \(0\text{,}\) full color = \(1\text{,}\) white = \(\infty\))

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  • Hue = argument (red = \(0\text{,}\) lime = \(\pii/2\text{,}\) cyan = \(\pii\text{,}\) purple = \(3\pii/2\)

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• Powers using De Moivre’s formula:

  \begin{equation*} (r\ee^{\ii\theta})^{n}=r^{n}\ee^{n\ii\theta} \end{equation*}

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• Roots of unity: \(z^n=1\) has roots at \(\ee^{2\pii\ii k/n}\) for \(k=0,1,\ldots,n-1\text{.}\)

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  We can see this in the domain coloring of \(z^n-1\text{,}\) where the colors cycle around the unit circle and approach black (zero) at the roots of unity.

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• Focus on \(f(z)=\dfrac{1}{z}\text{.}\)

  • Domain coloring: Swaps light and dark, and colors cycle the other way.

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• Division by zero and the Riemann sphere

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• Define a pole of a function as a zero of its reciprocal. But need to find a slightly different way to say this, as it’s missing some things.

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• Look at domain coloring plots of other complex functions, focusing on zeros and poles, perhaps as well as other stranger singularities (such as \(\ee^{-1/z}\)).

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