Univalent function: Difference between revisions
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==Comparison with real functions== |
==Comparison with real functions== |
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For [[real number|real]] [[analytic function]]s, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function |
For [[real number|real]] [[analytic function]]s, unlike for complex analytic (that is, holomorphic) functions, these statements fail to hold. For example, consider the function |
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:<math>f: (-1, 1) \to (-1, 1) \, </math> |
:<math>f: (-1, 1) \to (-1, 1) \, </math> |
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given by ''ƒ''(''x'') = ''x''<sup>3</sup>. 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). |
given by ''ƒ''(''x'') = ''x''<sup>3</sup>. 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). |
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== References== |
== References== |
Revision as of 23:19, 11 May 2013
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.
Examples
Any mapping of the open unit disc to itself, : where is univalent.
Basic properties
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 onto), 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
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 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). Consequently, if we enlarge the domain to an open subset G of the complex plane, it must fail to be one-to-one; 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
- 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.