By Steven G. Krantz

This is a e-book approximately complicated variables that provides the reader a brief and available advent to the main subject matters. whereas the assurance isn't accomplished, it definitely supplies the reader a superior grounding during this primary region. there are numerous figures and examples to demonstrate the vital principles, and the exposition is full of life and welcoming. An undergraduate desirous to have a primary examine this topic or a graduate pupil getting ready for the qualifying tests, will locate this booklet to be an invaluable source.

In addition to special principles from the Cauchy idea, the e-book additionally comprise sthe Riemann mapping theorem, harmonic services, the argument precept, basic conformal mapping and dozens of different crucial topics.

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

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**Additional resources for A Guide to Complex Variables**

**Example text**

46 CHAPTER 3. 5) open set and let f : U → C be holomorphic. Let the zero set of f be Z = {z ∈ U : f(z) = 0}. If there are a z0 ∈ Z and {zj }∞ j=1 ⊆ Z \ {z0 } such that zj → z0 , then f ≡ 0. Let us formulate the result in topological terms. We recall that a point z0 is said to be an accumulation point of a set Z if there is a sequence {zj } ⊆ Z \ {z0} with limj→∞ zj = z0. 5) open set U and if Z = {z ∈ U : f(z) = 0} has an accumulation point in U, then f ≡ 0. For the proof, suppose that the point 0 is an interior accumulation point of zeros {zj } of the holomorphic function f.

This explains why v, and hence F = u + iv, exists. The displayed theorem is false on a domain with a hole, such as an annulus. For example, the harmonic function u = log(x2 + y 2 ), defined on the annulus U = {z : 1 < |z| < 2}, has no harmonic conjugate on U. 4. 1 Real and Complex Line Integrals In this section we shall recast the line integral from calculus in complex notation. The result will be the complex line integral. The complex line integral is essential to the Cauchy theory, which we develop below, and that in turn is key to the argument principle and many of the other central ideas of the subject.

Then we say that p has a root of order 3 at 5, a root of order 8 at −2, and it has roots of order 1 at 3i and at −6. We also say that p has simple roots at 1 and −6. 5 Sequences of Holomorphic Functions and their Derivatives A sequence of functions gj defined on a common domain E is said to converge uniformly to a limit function g if, for each > 0, there is a number N > 0 such that for all j > N it holds that |gj (x) − g(x)| < for every x ∈ E. The key point is that the degree of closeness of gj (x) to g(x) is independent of x ∈ E.

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