Reality and Imagination

Grey and colorful.
Cutting through each other.
Chasing each other.
Meeting in the center,
leaning on each other,
forming an infinite line.

~

Reality and Imagination: Real and imaginary part of complex function 1/z:

~

The real part of 1/z is painted in shades of grey, the imaginary part in rainbow colors. Plots are created with a python script in SageMath: Each part is built from wire loops of constant real / imaginary part (parametric plots); colors change cyclically with function value.

If z is expressed as x + iy, real and imaginary parts of 1/z are obtained by multiplying nominator and denominator by x – iy. Moving away from the zero point, the denominator x2 + y2 finally wins, and both parts become zero. Denominators are x for the real part and -y for the imaginary part. At the zero point, the denominator goes to zero faster than the nominators, thus both parts have poles. As x and -y change sign when the zero point is crossed, each function is made up of two infinite towers, one extending to infinity, the other to minus infinity.

The asymmetric towers lean on each other, they meet in the vertical line erected at the zero point. x and -y are π/2 apart, speaking of them in terms of the complex plane. They chase each other, one lagging by π/2, like current and voltage in a resonant circuit, like force and amplitude.

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