Univalent function: Difference between revisions

Revision as of 22:34, 27 February 2013

Revision as of 23:19, 11 May 2013

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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'')&nbsp;=&nbsp;''x''³. This function is clearly one-to-one, however, its derivative is 0 at ''x''&nbsp;=&nbsp;0, and its inverse is not analytic, or even differentiable, on the whole interval&nbsp;(&minus;1,&nbsp;1). Consequently, if we enlarge the ___domain to an open subset <i>G</i> of the complex plane, it must fail to be one-to-one; and this is the case, since (for example) <i>f</i>(&epsilon;&omega;)&nbsp;= <i>f</i>(&epsilon;) (where &omega; is a [[primitive root of unity|primitive cube root of unity]] and &epsilon; is a positive real number smaller than the radius of <i>G</i> as a neighbourhood of 0.

== References==