Invitation to complex analysis

@inproceedings{Boas1987InvitationTC,
  title={Invitation to complex analysis},
  author={Ralph Philip Boas},
  year={1987}
}
Preface 1. From complex numbers to Cauchy's Theorem 2. Applications of Cauchy's Theorem 3. Analytic continuation 4. Harmonic functions and conformal mapping 5. Miscellaneous topics Index. 

The geometry of harmonic functions

(1994). The Geometry of Harmonic Functions. Mathematics Magazine: Vol. 67, No. 2, pp. 92-108.

Julius and Julia: Mastering the Art of the Schwarz Lemma

Abstract This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and

Boundary Schwarz inequalities arising from Rogosinski's lemma

We consider some Schwarz and Carathéodory inequalities at the boundary, as consequences of a lemma due to Rogosinski.

MASTERING THE ART OF THE SCHWARZ LEMMA

This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and the

Approaching Cauchy’s Theorem

Abstract We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is

2 00 4 On rational de nite summation

TLDR
The theoretical results proved provide an algorithm for computation of a large class of sums and a partial proof of van Hoeij-Abramov conjecture about the algorithmic possibility of computation of nite sums of rational functions.

The Cauchy Theory: A Fundamental Theorem

As with the theory of differentiation for complex-valued functions of a complex variable, the integration theory of such functions begins by mimicking and extending results from the theory for

Real analysis proof of Fundamental Theorem of Algebra using polynomial interlacing

The existence of a real quadratic polynomial factor, given any polynomial with real coefficients, is proven using only elementary real analysis. The aim is to provide an approachable proof to anyone

A characterization of holomorphic mappings on a poly-plane

More Eventual Positivity for Analytic Functions

Abstract Eventual positivity problems for real convergent Maclaurin series lead to density questions for sets of harmonic functions. These are solved for large classes of series, and in so doing,
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Apostol: Mathematical Analysis (Second Edition). Addison-Wesley Publishing Co

  • 1974

On the other hand, |z| R defines the closed ball B(0, R)

    By Liouville's theorem, g must be constant. Hence p must be constant