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Nonconformal Map

Let gamma be a path in C, w=f(z), and theta and phi be the tangents to the curves gamma and f(gamma) at z_0 and w_0. If there is an N such that

f^((N))(z_0)!=0
(1)
f^((n))(z_0)=0
(2)

for all n<N (or, equivalently, if f^'(z) has a zero of order N-1), then

f(z)=f(z_0)+(f^((N))(z_0))/(N!)(z-z_0)^N+(f^((N+1))(z_0))/((N+1)!)(z-z_0)^(N+1)+...
(3)
f(z)-f(z_0)=(z-z_0)^N[(f(N)(z_0))/(N!)+(f^((N+1))(z_0))/((N+1)!)(z-z_0)+...],
(4)

so the complex argument is

arg[f(z)-f(z_0)]=Narg(z-z_0)+arg[(f(N)(z_0))/(N!)+(f^((N+1))(z_0))/((N+1)!)(z-z_0)+...].
(5)

As z->z_0, arg(z-z_0)->theta and |arg[f(z)-f(z_0)]|->phi,

phi=Ntheta+arg[(f(N)(z_0))/(N!)]=Ntheta+arg[f(N)(z_0)].
(6)

See also

Conformal Mapping

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Cite this as:

Weisstein, Eric W. "Nonconformal Map." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NonconformalMap.html

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