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I'm not sure it's conformal transformation. He uses various complex functions
as solns. for Laplace's eq. log for cylinders linear for planes etc.
Maybe my understanding of the math. is rather weak, but it does seem more
direct than knowing a little known plane geometry fact, and appeal is also made
(note passive voice) to the log character of the potential.
They both appear
from ???, in one case a soln. try of an eq. known basically to describe
electrostatics and the other from geometry.
I agree it's difficult method, but not necessarily of limited applicability, as
I suspect a computer could iteratively solve for irregular shapes beginning
with one of high symmetry. (a la JD)
bc
P.s. DB left to the reader the math. manipulation at the end. I wonder, if as
in so many texts, this is the hardest part.