Geometric function theory: Difference between revisions

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==Univalent function==

Of special interest are those complex functions which are one-to-one. That is, for points <math>z_1</math>, <math>z_2</math>, in a ___domain <math>D</math>, they share a common value, <math>f(z_1)=f(z_2)</math> only if they are the same point <math>z_1=z_2</math>. A function <math>f</math> analytic in a ___domain <math>D</math> is said to be univalent there if it does not take the same value twice for all pairs of distinct points <math>z_1</math> and <math>z_2</math> in <math>D</math>, i.e <math>f(z_1) \neq f(z_2)</math> implies <math>z_1 \neq z_2</math>. Alternate terms in common use are ''schilichtschlicht''( this is german for plain, simple) and ''simple''. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.

==References==