Complex Variables

A Physical Approach with Applications

Steven G. Krantz

• Verlag: Chapman and Hall/CRC
• Jahr: 2019
• ISBN:
  9781000007183
• Auflage: 2
• Thema: Mathematics
• Thema: Applied Mathematics

The idea of complex numbers dates back at least 300 years—to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers.

This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications.

Features:

• This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysis

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• This book has an exceptionally large number of examples and a large number of figures.

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• The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking.

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• Incisive applications appear throughout the book.

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• Partial differential equations are used as a unifying theme.

Chapter 1

Basic Ideas

1.1 Complex Arithmetic

1.1.1 The Real Numbers

The real number system consists of both the rational numbers (numbers with terminating or repeating decimal expansions) and the irrational numbers (numbers with infinite, non-repeating decimal expansions). The real numbers are denoted by the symbol ℝ. We let ℝ² = {(x, y) : x ∈ ℝ, y ∈ ℝ} (Figure 1.1).

1.1.2 The Complex Numbers

The complex numbers ℂ consist of ℝ² equipped with some special algebraic operations. One defines

$\begin{array}{lll}\left(x,y\right)+\left(x^{\prime },y^{\prime }\right)\hfill & =\hfill & \left(x+x^{\prime },y+y^{\prime }\right),\hfill \\ \hfill \left(x,y\right)\cdot \left(x^{\prime },y^{\prime }\right)& =\hfill & \left(x{x}^{\prime }-y{y}^{\prime },x{x}^{\prime }+y{x}^{\prime }\right).\hfill \end{array}$

These operations of + and · are commutative and associative.

EXAMPLE 1 We may calculate that

$\left(3,7\right)+\left(2,-4\right)=\left(3+2,7+\left(-4\right)\right)=\left(5,3\right).$

Also

$\left(3,7\right)\cdot \left(2,-4\right)=\left(3\cdot 2-7\cdot \left(-4\right),3\cdot \left(-4\right)+7\cdot 2\right)=\left(34,2\right).$

Figure 1.1: A point in the plane.

□

Of course we sometimes wish to subtract complex numbers. We define

$z-w=z+\left(-w\right).$

Thus if z = (11, −6) and w = (1, 4) then

$z-w=z+\left(-w\right)=\left(11,-6\right)+\left(-1,-4\right)=\left(10,-10\right).$

We denote (1, 0) by 1 and (0, 1) by i. We also denote (0, 0) by 0. If α ∈ ℝ, then we identify α with the complex number (α, 0). Using this notation, we see that

$\alpha \cdot \left(x,y\right)=\left(\alpha ,0\right)\cdot \left(x,y\right)=\left(\alpha x,\alpha y\right).$

(1.1)

In particular,

$1\cdot \left(x,y\right)=\left(1,0\right)\cdot \left(x,y\right)=\left(x,y\right).$

We may calculate that

$x\cdot 1+y\cdot i=\left(x,0\right)\cdot \left(1,0\right)+\left(y,0\right)\cdot \left(0,1\right)=\left(x,0\right)+\left(0,y\right)=\left(x,y\right).$

(1.2)

Thus every complex number (x, y) can be written in one and only one fashion in the form x · 1 + y · i with x, y ∈ ℝ. We usually write the number even more succinctly as x + iy.

EXAMPLE 2 The complex number (−2, 5) is usually written as

$\left(-2,5\right)=-2+5i.$

The complex number (4, 9) is usually written as

$\left(4,9\right)=4+9i.$

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