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.
Elaboration
In the above figure, consider D
1 and D
2 as two simply connected regions different from 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-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 ℂ can be mapped conformally onto each other."
Univalent Function
We know that a complex function is a multiple valued function. That is, for distinct points z
1, z
2,... in a domain D, they may share a common value, f(z
1)=f(z
2)=... But if we restrict a complex function to be injective( one-one ), then we obtain a class of functions, viz, univalent functions. 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)≠f(z
2) implies z
1≠z
2. Alternate terms in common use are schilicht and simple. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.
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.
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