Univalent function

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 \phi_a of the open unit disc to itself. The function

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

where |a|<1, is univalent.

Basic properties[edit]

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

f: G \to \Omega

is a univalent function such that f(G) = \Omega (that is, f is surjective), 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.

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

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

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).

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.