Conformal Map

A Transformation which preserves Angles is known as conformal. For a transformation to be conformal, it must be an Analytic Function and have a Nonzero Derivative. Let $\theta$ and $\phi$ be the tangents to the curves $\gamma$ and $f(\gamma)$ at $z_0$ and $w_0$,

$$w-w_0 \equiv f(z)-f(z_0) = {f(z)-f(z_0)\over z-z_0} (z-z_0)$$ (1)

$$\arg(w-w_0) = \arg\left[{f(z)-f(z_0)\over z-z_0}\right]+\arg(z-z_0).$$ (2)

Then as $w\to w_0$ and $z\to z_0$,

$$\phi = \arg f'(z_0)+\theta$$ (3)

$$\vert w\vert = \vert f'(z_0)\vert \,\vert z\vert.$$ (4)

See also Analytic Function, Harmonic Function, Möbius Transformation, Quasiconformal Map, Similar

References

Arfken, G. ``Conformal Mapping.'' §6.7 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 392-394, 1985.

Bergman, S. The Kernel Function and Conformal Mapping. New York: Amer. Math. Soc., 1950.

Katznelson, Y. An Introduction to Harmonic Analysis. New York: Dover, 1976.

Morse, P. M. and Feshbach, H. ``Conformal Mapping.'' §4.7 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 358-362 and 443-453, 1953.

Nehari, Z. Conformal Map. New York: Dover, 1982.