In the real numbers, we can express a relationship between two sets of numbers by mapping elements from one set to elements in another. We call this mapping a function.
To visualize this, we can plot out a graph of y against x, where y = f(x).
However, this is not possible with complex numbers, which are represented with a 2-dimensional plane. Thus, we map a complex number from one Argand plane to another Argand plane. By seeing how each function transforms a square array of points (red to blue) and the unit circle (black), we can reveal some very interesting properties of how complex numbers behave under a function.
The available functions are as follows:
- Exponential (
w = e^z)
- Sine (
w = sin(z))
- Cosine (
w = cos(z))
- Tangent (
w = tan(z))
- Logarithmic (
w = ln(z))
- Power (
w = z^n)
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| Exponential Mapping |
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| Sine Mapping |
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| Cosine Mapping |
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| Tangent Mapping |
The map for the tangent function is not very clear when represented with points, so here's an image of the mapping of a square grid with a tangent function (image courtesy of Wolfram).
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| Logarithmic Mapping |
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| Power Mapping (n = 2) |
To explore how the complex plane transforms under these functions, you can check out the simulation
here.
