Univalent function

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In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is one-to-one.

Contents

[edit] Examples

Any mapping φa of the open unit disc to itself,

\phi_a(z) =\frac{z-a}{1 - \bar{a}z},

where |a|\le 1, is univalent.

[edit] Basic properties

One can prove that if G and Ω are two open connected sets in the complex plane, and

f: G \to \Omega

is a univalent function such that f(G) = Ω (that is, f is onto), then the derivative of f is never zero, f is invertible, and its inverse f - 1 is also holomorphic. More, one has by the chain rule

(f^{-1})'(f(z)) = \frac{1}{f'(z)}

for all z in G.

[edit] Comparison with real functions

For real analytic functions, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function

f: (-1, 1) \to (-1, 1)

given by f(x) = x3. This function is clearly one-to-one, however, its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval ( - 1,1).

[edit] References

  • John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, 1978. ISBN 0-387-90328-3.
  • John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, 1996. ISBN 0-387-94460-5.

This article incorporates material from univalent analytic function on PlanetMath, which is licensed under the GFDL.

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