Area of the unit disk after a Möbius transformation

Let f(z) = (az + b)/(cz + d) where = ad - bc 1.

Iff has no singularity inside the unit disk, i.e. if |d/c| > 1, then the image of the unit disk under f is another disk. What is the area of that disk?

The calculation is complicated, but the result turns out to be

Area = ||^2 / (|d|^2 - |c|^2)^2.

Just as a sanity check, set c = 0 andd = 1. Then we multiply the disk bya and shift byb. The shift doesn't change the area, and multiplying bya multiples the area by |a|^2, which is consistent with our result.

As another sanity check, note that the area is infinite ifc =d, which is correct since there would be a singularity atz = -1.

Finally, here's a third sanity check in the form of Python code.

from numpy import linspace, pi, expa, b, c, d = 9, 15j, 20, 25jtheory_r = abs(a*d - b*c)/(abs(d)**2 - abs(c)**2)print("theory r:", theory_r)t = linspace(0, 2*pi, 10000)z = exp(1j*t)w = (a*z + b)/(c*z + d)approx_r = (max(w.real) - min(w.real))/2print("approx r:", approx_r)

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