Möbius Transformation

Let a in C and |a|<1, then


is a Möbius transformation, where a^_ is the complex conjugate of a. phi_a is a conformal mapping self-map of the unit disk D for each a, and specifically of the boundary of the unit disk to itself. The same holds for (phi_a)^(-1)=phi_(-a).

Any conformal self-map of the unit disk to itself is a composition of a Möbius transformation with a rotation, and any conformal self-map f of the unit disk can be written in the form


for some Möbius transformation phi_b and some complex number w with |w|=1 (Krantz 1999, p. 81).

See also

Kleinian Group, Linear Fractional Transformation

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Krantz, S. G. "Möbius Transformations." §6.2.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 81, 1999.Needham, T. "Möbius Transformations and Inversion." Ch. 3 in Visual Complex Analysis. New York: Clarendon Press, pp. 122-188, 2000.

Referenced on Wolfram|Alpha

Möbius Transformation

Cite this as:

Weisstein, Eric W. "Möbius Transformation." From MathWorld--A Wolfram Web Resource.

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