Calculus with Complex Numbers

John B. Reade

• Publisher: CRC Press
• Year: 2003
• Print ISBN: 9780367834517
• eBook ISBN: 9781134394418
• Edition: 1
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This practical treatment explains the applications complex calculus without requiring the rigor of a real analysis background. The author explores algebraic and geometric aspects of complex numbers, differentiation, contour integration, finite and infinite real integrals, summation of series, and the fundamental theorem of algebra. The Residue Theo

Chapter 1
Complex numbers

1.1 The square root of minus one

Complex numbers originate from a desire to extract square roots of negative numbers. They were first taken seriously in the eighteenth century by mathematicians such as de Moivre, who proved the first theorem in the subject in 1722. Also Euler, who introduced the notation i for

, and who discovered the mysterious formula e^iθ = cosθ + i sin θ in 1748. And third Gauss, who was the first to prove the fundamental theorem of algebra concerning existence of roots of polynomial equations in 1799. The nineteenth century saw the construction of the first model for the complex numbers by Argand in 1806, later known as the Argand diagram, and more recently as the complex plane. Also the first attempts to do calculus with complex numbers by Cauchy in 1825. Complex numbers were first so called by Gauss in 1831. Previously they were known as imaginary numbers, or impossible numbers. It was not until the twentieth century that complex numbers found application to science and technology, particularly to electrical engineering and fluid dynamics.

If we want square roots of negative numbers it is enough to introduce

since then, for example,

. Combining i with real numbers by addition and multiplication cannot produce anything more general than x + iy where x, y are real. This is because the sum and product of any two numbers of this form are also of this form. For example,

Subtraction produces nothing new since, for example,

Neither does division since, for example,

The number 3 — 4i is called the conjugate of 3 + 4i. For any x + iy we have

so division can always be done except when x = y = 0, that is, when x + iy = 0.

It is also possible to extract square roots of numbers of the form x + iy as numbers of the same form. For example, suppose

then we have

So we require

The second equation gives B = 1/A, which on sub...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Chapter 1 Complex numbers
• Chapter 2 Complex functions
• Chapter 3 Derivatives
• Chapter 4 Integrals
• Chapter 5 Evaluation of finite real integrals
• Chapter 6 Evaluation of infinite real integrals
• Chapter 7 Summation of series
• Chapter 8 Fundamental theorem of algebra
• Solutions to examples
• Appendix 1: Cauchy’s theorem
• Appendix 2: Half residue theorem
• Bibliography