This page is a sub-page of our page on Calculus.
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The sub-pages of this page are:
• Complex Derivative
• Complex trigonometry
• Conformal Mapping
• Inversion
• Möbius transformations
• Stereographic Projection
• The Riemann Zeta function
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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 interactive simulation that created this movie.
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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• Maxwell and Dirac theories as an already unified theory
Conceptual background:
Historical background:
• The Evolution Of Geometric Arithmetic
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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):
Interactive simulation of the devil transformed by exp(z).
Complex trigonometric functions
Devil transformed by complex sin: sin(z+p) for different values of p (moving red dot):
Interactive simulation of the devil transformed by sin(z).
Angels and Devils transformed by complex sin: