By Steven G. Krantz

This is a publication approximately advanced variables that offers the reader a short and obtainable advent to the main subject matters. whereas the insurance isn't entire, it definitely offers the reader an outstanding grounding during this primary region. there are various figures and examples to demonstrate the imperative rules, and the exposition is vigorous 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 publication to be an invaluable source.

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

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

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The end of the last proof bears some commentary. We prove that ∂f/∂z ≡ 0. But we know, since f is holomorphic, that ∂f/∂z ≡ 0. It follows from linear algebra that ∂f/∂x ≡ 0 and ∂f/∂y ≡ 0. Then calculus tells us that f is constant. The reasoning that establishes Liouville’s theorem can also be used to prove this more general fact: If f : C → C is an entire function and if for some real number C and some positive integer k, it holds that |f(z)| ≤ C · (1 + |z|)k for all z, then f is a polynomial in z of degree at most k.

3). 38 Chapter 2. 1 A Formula for the Derivative Let U ⊆ C be an open set and let f be holomorphic on U. Then f ∈ C ∞ (U). Moreover, if D(P, r) ⊆ U and z ∈ D(P, r), then ∂ ∂z k f(z) = k! 2πi |ζ−P |=r f(ζ) dζ, (ζ − z)k+1 k = 0, 1, 2, . . 1) under the integral sign. 2 The Cauchy Estimates If f is a holomorphic on a region containing the closed disc D(P, r) and if |f| ≤ M on D(P, r), then ∂k M · k! f(P ) ≤ . 1). 39 40 CHAPTER 3. , f : C → C is holomorphic. For instance, any holomorphic polynomial is entire, ez is entire, and sin z, cos z are entire.

Zn and w1, . . , wn are complex numbers, then 2 n z j wj j=1 n ≤ j=1 n |zj | 2 · j=1 |wj |2 . 3) This result is immediate from the Triangle Inequality: Just square both sides and multiply everything out. 1 Holomorphic Functions Continuously Differentiable and C k Functions In this book we will frequently refer to a domain or a region U ⊆ C. 5). Holomorphic functions are a generalization of complex polynomials. But they are more flexible objects than polynomials. The collection of all polynomials is closed under addition and multiplication.