Geometric function theory

Geometric function theory is the study of geometric properties of analytic functions. A fundamental result in the theory is the Riemann mapping theorem.

Riemann mapping theorem

Let $z_{0}$ be a point in a simply-connected region $D_{1}(D_{1}\neq \mathbb {C} )$ and $D_{1}$ having at least two boundary points. Then there exists a unique analytic function $w=f(z)$ mapping $D_{1}$ bijectively into the open unit disk $|w|<1$ such that $f(z_{0})=0$ and $f'(z_{0})>0$.

It should be noted that while Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function.

Elaboration

In the above figure, consider $D_{1}$ and $D_{2}$ as two simply connected regions different from $\mathbb {C}$. The Riemann mapping theorem provides the existence of $w=f(z)$ mapping $D_{1}$ onto the unit disk and existence of $w=g(z)$ mapping $D_{2}$ onto the unit disk. Thus $g^{-1}f$ is a one-to-one mapping of $D_{1}$ onto $D_{2}$. If we can show that $g^{-1}$, and consequently the composition, is analytic, we then have a conformal mapping of $D_{1}$ onto $D_{2}$, proving "any two simply connected regions different from the whole plane $\mathbb {C}$ can be mapped conformally onto each other."

Univalent function

Of special interest are those complex functions which are one-to-one. That is, for points $z_{1}$, $z_{2}$, in a domain $D$, they share a common value, $f(z_{1})=f(z_{2})$ only if they are the same point $z_{1}=z_{2}$. A function $f$ analytic in a domain $D$ is said to be univalent there if it does not take the same value twice for all pairs of distinct points $z_{1}$ and $z_{2}$ in $D$, i.e. $f(z_{1})\neq f(z_{2})$ implies $z_{1}\neq z_{2}$. Alternate terms in common use are schlicht( this is German for plain, simple) and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.

References

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