Univalent function explained

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

The function

f\colonz\mapsto2z+z2

is univalent in the open unit disc, as

f(z)=f(w)

implies that

f(z)-f(w)=(z-w)(z+w+2)=0

. As the second factor is non-zero in the open unit disc,

z=w

so

f

is injective.

Basic properties

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))=

1
f'(z)

for all

z

in

G.

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

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

given by

f(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(\varepsilon\omega)=f(\varepsilon)

(where

\omega

is a primitive cube root of unity and

\varepsilon

is a positive real number smaller than the radius of

G

as a neighbourhood of

0

).

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Univalent function".

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