Complex Variables I Essentials

By Alan D. Solomon

REA’s Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Complex Variables I includes functions of a complex variable, elementary complex functions, integrals of complex functions in the complex plane, sequences and series, and poles and residues.

• Language: English
• Publisher: Research & Education Association
• Release date: Jan 1, 2013
• ISBN: 9780738672434

Book preview

RESIDUES

CHAPTER 1

BASIC DEFINITIONS

1.1 COMPLEX NUMBERS

Natural numbers are the counting numbers 1, 2, 3, ... .

Negative integers are the numbers–1,–2,–3, ... .

Intergers are the natural numbers, negative integers, and zero.

Fractions, or rational numbers, are ratios of integers, 1/2, 7/8, —5/6, or generally, m/n, for n ≠ 0.

Irrational numbers are those that are not rational, such as √2, π, e,√π, etc.

Real numbers are all the rational and irrational numbers.

The imaginary unit or complex unit i is defined by

(1.1)

Complex numbers are of the form

(1.2)

with x, y real; x is the real part of z, and y, the imaginary part,

(1.3)

Examples are 1 + 2i, —3i, 0 ,√2 + 7i, etc.

The complex conjugate z– of z is

(1.4)

1.2 ARITHMETIC OF THE COMPLEX NUMBERS

For z1 = x1 + iy1, z2 = x2 + iy2 we define:

Addition: Their sum is

(1.5)

Subtraction: Their difference is

(1.6)

Multiplication: Their product is

(1.7)

A special case is the product of a number by its complex conjugate

(1.8)

which is always a positive number, vanishing only when z = 0. For z2, ≠ 0 we define division by the quotient (1.9)

(1.9)

In terms of real and imaginary parts,

A special case is the reciprocal, or multiplicative inverse of the number z ≠ 0,

(1.10)

1.3 COMMUTATIVITY, ASSOCIATIVITY, AND DISTRIBUTIVITY

Addition and multiplication are commutative; for any z1, z2,

(1.11)

(1.12)

Addition and multiplication are associative: for any z1,z2, z3,

(1.13)

(1.14)

Multiplication is distributive with respect to addition (distributivity),

(1.15)

for any z1, z2, z3.

1.4 THE COMPLEX PLANE

There is a one-to-one correspondence between the complex numbers z = x +iy and the points (x,y) of the x - y plane (Figure 1.1).

Fig 1.1 Correspondence between numbers z = x + iy and points (x,y)

10. The complex plane or Argand diagram is the x, y plane viewed as the plane of the complex numbers z = x + iy.

11. The x and y - axes of the complex plane are called the real and imaginary axes.

The parallelogram law states that z1 + z2 is the fourth corner of a parallelogram defined by the origin z = 0 and the end points z1, z2 (Figure 1.2).

Fig 1.2Parallelogram Law

of z is the reflection of z in the real axis (Figure 1.3).

Figure 1.3The Complex Conjugate; Reflection Across the Real Axis

A real multiple cz of a complex number z is represented by a point on the line from the origin z = 0 to z; if c > 0 then it lies on the same side of the origin; if c < 0 it lies on the opposite side (Figure 1.4).

Fig 1.4ac zforc> 0

Fig 1.4bc zforc< 0

1.5 THE MODULUS OF z

12. The modulus or absolute value of z = x + iy, denoted by mod z or |z | is

(1.16)

The modulus obeys the usual rules of the absolute value of real numbers, including the product relation

(1.17)

and the triangle inequality (Figure 1.5)

(1.18)

Figure 1.5The Triangle Inequality, |z1+z2| ≤ |z1| + |z2|

with equality holding only if z1 is a real multiple of z2.

The triangle inequality holds for the sum of any number of values,

(1.19)

|z | is the distance from z to the origin (Figure 1.6).

Figure 1.6The modulus is the distance to the origin.

|z1 — z2 | is the distance between the points z1,z2 (Figure 1.7).

Fig 1.7|z1—z2| is the distance betweenz1, z2.

An important implication of the triangle inequality is the inequality (Figure 1.8)

(1.20)

Figure 1.8 ||z2| —