A Guide to Complex Variables by Steven G. Krantz

By Steven G. Krantz

This is a ebook approximately complicated variables that offers the reader a brief and obtainable creation to the main issues. whereas the insurance isn't really entire, it definitely offers the reader a superior grounding during this basic region. there are lots of figures and examples to demonstrate the important principles, and the exposition is full of life and alluring. An undergraduate eager to have a primary examine this topic or a graduate pupil getting ready for the qualifying assessments, will locate this ebook to be an invaluable source.

In addition to big principles from the Cauchy conception, the e-book additionally contain sthe Riemann mapping theorem, harmonic features, the argument precept, common conformal mapping and dozens of different important topics.

Readers will locate this ebook to be an invaluable spouse to extra exhaustive texts within the box. it's a worthy source for mathematicians and non-mathematicians alike.

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Conformality” is an important geometric property of holomorphic functions that make these functions useful for modeling incompressible fluid flow and other physical phenomena. 5. See also [KRA6]. 4. 1 Holomorphic and Harmonic Functions Harmonic Functions A C 2 function u is said to be harmonic if it satisfies the equation ∂2 ∂2 + ∂x2 ∂y 2 u = 0. 1) This equation is called Laplace’s equation, and is frequently abbreviated as u = 0. 2) How They are Related If f is a holomorphic function and f = u + iv is the expression of f in terms of its real and imaginary parts, then both u and v are harmonic.

Holomorphic functions enjoy both properties. Now we shall discuss them in detail. Let f be holomorphic in a neighborhood of P ∈ C. Let w1, w2 be complex numbers of unit modulus. 1) f(P + tw2 ) − f(P ) . 3) |Dw1 f(P )| = |Dw2 f(P )| . 30 Chapter 2. 4) If |f (P )| = 0, then the directed angle from w1 to w2 equals the directed angle from Dw1 f(P ) to Dw2 f(P ). 4) holds at P , then f has a complex derivative at P . 3) holds at P , then either f or f has a complex derivative at P . Thus a function that is conformal (in either sense) at all points of an open set U must possess the complex derivative at each point of U.

It then follows from an easy connectedness argument (more on this below) that f ≡ 0. 1. A set S is said to be discrete if for each s ∈ S there is an > 0 such that D(s, ) ∩ S = {s}. 2. 2 thus asserts that if f is a non-constant holomorphic function on a connected open set, then its zero set is discrete or, less formally, the zeros of f are isolated. 2 does not rule out the possibility that the zero set of f can have accumulation points in C \ U; in particular, a non-constant holomorphic function on an open set U can indeed have zeros accumulating at a point of ∂U.

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