By Steven G. Krantz

This is a booklet approximately advanced variables that provides the reader a brief and obtainable creation to the most important subject matters. whereas the insurance isn't really entire, it definitely supplies the reader an outstanding grounding during this primary region. there are numerous figures and examples to demonstrate the crucial principles, and the exposition is vigorous and alluring. An undergraduate desirous to have a primary examine this topic or a graduate scholar getting ready for the qualifying checks, will locate this ebook to be an invaluable source.

In addition to special principles from the Cauchy conception, the e-book additionally contain sthe Riemann mapping theorem, harmonic capabilities, the argument precept, normal conformal mapping and dozens of alternative critical topics.

Readers will locate this e-book to be an invaluable better half to extra exhaustive texts within the box. it's a important source for mathematicians and non-mathematicians alike.

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

**Example text**

Take the center of the disc in the Cauchy formula to be the origin 0. 2. The Zeros of a Holomorphic Function Cauchy formula as f(z) = = 1 2πi f(ζ) dζ ∂D(0,r) ζ − z 1 1 f(ζ) · · ζ 1 − z/ζ ∂D(0,r) ∂D(0,r) f(ζ) · 1 · ζ ∂D(0,r) f(ζ) · ζ j+1 = = = j 1 ∞ dζ (z/ζ)j dζ j=0 dζ · z j aj · z j . We see explicitly that aj = ∂D(0,r) f(ζ) · 1 ζ j+1 dζ , and this corresponds, by the Cauchy formula, to derivatives of f. Of course the series converges absolutely and uniformly for |z| < r = |ζ|. The key point here is that holomorphic functions are analytic because the Cauchy kernel is analytic.

If instead f satisfies ∂f/∂z ≡ 0 on an open set U, then f does not depend on z (but it does depend on z). 4. 4 Definition of Holomorphic Function Functions f that satisfy (∂/∂z)f ≡ 0 are the main concern of complex analysis. 1) at every point of U. 2). To see this, we calculate: ∂ f(z) ∂z 1 ∂ ∂ = +i [u(z) + iv(z)] 2 ∂x ∂y ∂u ∂v ∂u ∂v = − + i + . 2) Of course the far right-hand side cannot be identically zero unless each of its real and imaginary parts is identically zero. It follows that ∂u ∂v − =0 ∂x ∂y 18 CHAPTER 1.

Thus γ = {γ(t) : t ∈ [a, b]} . Often we follow the custom of referring to either the function or the image with the single symbol γ. It will be clear from context what is meant. 1. It is often convenient to write γ(t) = (γ1 (t), γ2(t)) or γ(t) = γ1 (t) + iγ2(t). 1) For example, γ(t) = (cos t, sin t) = cos t + i sin t, t ∈ [0, 2π], describes the unit circle in the plane. The circle is traversed in a counterclockwise manner as t increases from 0 to 2π. 1. 23 24 CHAPTER 2. 1: Curves in the plane. 2 Closed Curves The curve γ : [a, b] → C is called closed if γ(a) = γ(b).