## Geometric Function Theory, Explorations in Complex Analysis

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Cancel Save. For example, they can look like a sphere or a torus or several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions , or the logarithm. A holomorphic function on an open subset of the complex plane is called univalent if it is injective.

More, one has by the chain rule. 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. Although Riemann's mapping theorem demonstrates the existence of a mapping function, it does not actually exhibit this function. An example is given below. The Schwarz lemma , named after Hermann Amandus Schwarz , is a result in complex analysis about holomorphic functions from the open unit disk to itself.

The lemma is less celebrated than stronger theorems, such as the Riemann mapping theorem , which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions. Schwarz Lemma.

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The maximum principle is a property of solutions to certain partial differential equations , of the elliptic and parabolic types. Roughly speaking, it says that the maximum of a function in a domain is to be found on the boundary of that domain. Specifically, the strong maximum principle says that if a function achieves its maximum in the interior of the domain, the function is uniformly a constant. The weak maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces which is its origin and algebraic curves.

Imaginary Numbers Are Real [Part 1: Introduction]

In the case of an unramified covering map of surfaces. What the Riemann—Hurwitz formula does is to add in a correction to allow for ramification sheets coming together. The number n is called the ramification index at P and also denoted by e P. Therefore, we find a "corrected" formula. This formula is known as the Riemann—Hurwitz formula and also as Hurwitz's theorem.

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Main article: Riemann surface. Main article: Univalent function. Main article: Schwarz lemma.

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