# Calculus of One Complex Variable

This page is a sub-page of our page on Calculus.

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Related KMR-pages:

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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The complex exponential function

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A planar electro-magnetic wave:

The electric part of the wave: $\, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \,(\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, \omega \, t)} \,$

The magnetic part of the wave: $\, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, e^{ \, i \, (\mathbf{\hat{k}} \cdot \mathbf{x} \, - \, (\omega \, + \, \pi/2) \, t)} \,$

The entire wave: $\, E_m(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, = \, E(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \, + \, B(\mathbf{\hat{k}}, \mathbf{x}, \omega, t) \,$

Its Poynting vector : $\, S \, = \, \frac{1}{{\mu}_0} \, E \, \times \, B$

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Conceptual background:

Historical background:

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Angels and devils: exp(z + p) for different values of p (moving black dot):

Devil transformed by exp(z+p) for different values of p (moving red dot):

Complex trigonometric functions

Devil transformed by complex sin: sin(z+p) for different values of p (moving red dot):

Angels and Devils transformed by complex sin: