Triplet state

2013-07-12T07:28:10Z

203.250.76.27: /* Two spin-1/2 particles */

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 function|injective]].

== Examples==
Any mapping <math>\phi_a</math> of the open [[unit disc]] to itself, :<math>\phi_a(z) =\frac{z-a}{1 - \bar{a}z},</math> where <math>|a|<1,</math> is univalent.

==Basic properties==
One can prove that if <math>G</math> and <math>\Omega</math> are two open [[connected space|connected]] sets in the complex plane, and

:<math>f: G \to \Omega</math>

is a univalent function such that <math>f(G) = \Omega</math> (that is, <math>f</math> is [[Surjective_function|surjective]]), then the derivative of <math>f</math> is never zero, <math>f</math> is [[invertible]], and its inverse <math>f^{-1}</math> is also holomorphic. More, one has by the [[chain rule]]

:<math>(f^{-1})'(f(z)) = \frac{1}{f'(z)}</math>

for all <math>z</math> in <math>G.</math>

== Comparison with real functions ==

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

:<math>f: (-1, 1) \to (-1, 1) \, </math>

given by ''&fnof;''(''x'') = ''x''<sup>3</sup>. 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 root of unity|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.

{{PlanetMath attribution|title=univalent analytic function|id=5633}}

[[Category:Analytic functions]]

[[is:Eintæk vörpun]]

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Triplet state