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proof of conformal mapping theorem
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(Proof)
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Let
be a domain, and let
be an analytic function. By identifying the complex plane
with
, we can view as a function from
to itself:
with and real functions. The Jacobian matrix of is
As an analytic function, satisfies the Cauchy-Riemann equations, so that and . At a fixed point
, we can therefore define
and
. We write in polar coordinates as
and get
Now we consider two smooth curves through , which we parametrize by
and
. We can choose the parametrization such that
. The images of these curves under are
and
, respectively, and their derivatives at are
and, similarly,
by the chain rule. We see that if
, transforms the tangent vectors to and at (and therefore in ) by the orthogonal matrix
and scales them by a factor of . In particular, the transformation by an orthogonal matrix implies that the angle between the tangent vectors is preserved. Since the determinant of is 1, the transformation also preserves orientation (the direction of the angle between the tangent vectors). We conclude that is a conformal mapping at each point where its derivative is nonzero.
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"proof of conformal mapping theorem" is owned by pbruin.
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(view preamble)
| Keywords: |
analytic function, conformal mapping |
This object's parent.
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Cross-references: point, conformal mapping, orientation, preserves, determinant, angle, implies, transformation, factor, orthogonal matrix, tangent vectors, Transforms, chain rule, derivatives, images, curves, smooth, polar coordinates, fixed point, Cauchy-Riemann equations, Jacobian matrix, real functions, function, complex plane, analytic function, domain
This is version 4 of proof of conformal mapping theorem, born on 2003-07-23, modified 2004-04-14.
Object id is 4502, canonical name is ProofOfConformalMappingTheorem.
Accessed 2211 times total.
Classification:
| AMS MSC: | 30C35 (Functions of a complex variable :: Geometric function theory :: General theory of conformal mappings) |
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Pending Errata and Addenda
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