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≠ 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 Re 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 exhibits this function.

References

• Krantz, Steven (2006). Geometric Function Theory: Explorations in Complex Analysis. Springer. ISBN 0817643397.
• Noor, K.I. Lecture notes on Introduction to Univalent Functions. CIIT, Islamabad, Pakistan.