complex analysis | Christopher Olah's Blog

Posts Tagged ‘complex analysis’

The Real 3D Mandelbrot Set

August 8, 2011

Perhaps more than any other area of serious mathematics, fractals and more specifically the Mandelbrot set, have attracted a great deal of public interest. Continuing this pattern, the Mandelbulb set caused a great deal of excitement. As the Mandelbulb website it says, however, “there’s good reason to believe that it isn’t the real McCoy.” And so the question is left hanging, what is the real 3D Mandelbrot Set? In this essay, I will present the fractal that I believe deserves this position and why this is the case. (more…)

Tags:complex analysis, complex dynamics, fractals, mandelbrot set
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Complex Powers

March 9, 2010

Exponents + Complex Numbers = Cool Stuff

I’ve been messing around with these sort of things and discovering all sorts of neat stuff. I haven’t got far enough in Needham to really understand all of it, but here are some things I have found.

The plots of $x^k$ where $k$ is a point on the complex unit circle are fascinating. Here’s an animation I made:

You can also watch a high-quality video on youtube.

At $k=i$ argument and magnitude of the output seem to be switched. They swirl into these circles and then swirl out such that the high magnitude and low magnitude ends are switched.

Another interesting thing is that $i^x$ and $e^{ix}$ are related. In fact: $i^{-2ix/\pi}=e^x$ … $i^{-2i/\pi}=e$

Update: The reason why this happens is fairly straight forward. Consider $x^{e^{\pi i n}} = (re^{i\theta})^{\cos(\pi n)+i\sin(\pi n)}=e^{\ln(r)\cos(\pi n)}*e^{\ln(r)i\sin(\pi n)}*e^{i\theta\cos(\pi n)}*e^{-\theta\sin(\pi n)}=e^{\ln(r)\cos(\pi n)-\theta\sin(\pi n)}*e^{i\theta\cos(\pi n)+\ln(r)i\sin(\pi n)}$ and thus magnitude and argument switch as $n$ increases.