Let f(z) = (az + b)/(cz + d) where Δ = ad − bc ≠ 1.
If f 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 = π |Δ|² / (|d|² − |c|²)².
Just as a sanity check, set c = 0 and d = 1. Then we multiply the disk by a and shift by b. The shift doesn’t change the area, and multiplying by a multiples the area by |a|², which is consistent with our result.
As another sanity check, note that the area is infinite if c = d, which is correct since there would be a singularity at z = −1.
Finally, here’s a third sanity check in the form of Python code.
from numpy import linspace, pi, exp
a, b, c, d = 9, 15j, 20, 25j
theory_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))/2
print("approx r:", approx_r)