Complex Analysis, Dennis G. Zill, Patrick D. Shanahan

Complex Analysis, Dennis G. Zill, Patrick D. Shanaha

 

    In this book, Each chapter begins with its own opening page that includes a table of contents and a brief introduction describing the material to be covered in the chapter. Moreover, each section in a chapter starts with introductory comments on the specifics covered in that section.

Here is a brief description of the topics covered in the seven chapters. • 

• Chapter 1 The complex number system and the complex plane are examined in detail.
• Chapter 2 Functions of a complex variable, limits, continuity, and mappings are introduced.
• Chapter 3 The all-important concepts of the derivative of a complex function and analyticity of a function are presented.
• Chapter 4 The trigonometric, exponential, hyperbolic, and logarithmic functions are covered. The subtle notions of multiple-valued functions and branches are also discussed.
• Chapter 5 The chapter begins with a review of real integrals (including line integrals). The definitions of real line integrals are used to motivate the definition of the complex integral. The famous CauchyGoursat theorem and the Cauchy integral formulas are introduced in this chapter. Although we use Green’s theorem to prove Cauchy’s theorem, a sketch of the proof of Goursat’s version of this same theorem is given in an appendix.
• Chapter 6 This Chapter introduce the concepts of complex sequences and infinite series. The focus of the chapter is on Laurent series, residues, and the residue theorem. Evaluation of complex as well as real integrals, summation of infinite series, and calculation of inverse Laplace and inverse Fourier transforms are some of the applications of residue theory that are covered.
• Chapter 7 Complex mappings that are conformal are defined and used to solve certain problems involving Laplace’s partial differential equation.

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