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 injective.

Examples[edit]

Any mapping of the open unit disc to itself. The function

where is univalent.

Basic properties[edit]

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

is a univalent function such that (that is, is surjective), then the derivative of is never zero, is invertible, and its inverse is also holomorphic. More, one has by the chain rule

for all in

Comparison with real functions[edit]

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

given by ƒ(x) = x3. This function is clearly injective, but its derivative is 0 at x = 0, and its inverse is not analytic, or even differentiable, on the whole interval (−1, 1). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be injective; and this is the case, since (for example) f(εω) = f(ε) (where ω is a primitive cube root of unity and ε is a positive real number smaller than the radius of G as a neighbourhood of 0).

See also[edit]

References[edit]

  • 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 Creative Commons Attribution/Share-Alike License.