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Methods of Mathematical Physics, Third Edition by Bertha Jeffreys, Harold Jeffreys

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Chapter 13CONFORMAL REPRESENTATION

Plus ça change, plus c’est la même chose.

ALPHONSE KABR, Les Guépes, 1849

13·01. Conditions for conformai mapping.

Let ζ and z be two complex variables related by

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where f(ζ) is an analytic function. Then if ζ moves along a curve in the plane of ξ, η, z will move along a curve in the plane of x, y. Every value of ζ that f(ζ) is defined for will identify a point in the x, y plane, and conversely if the inverse function exists for a value of z a value of ζ will be identified. Thus (1) can be regarded as a transformation, enabling us to map at least a part of the ξ, η plane on the x, y plane and conversely.

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