Luis Barreira - Complex Analysis and Differential Equations

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Part 1
Complex Analysis

Luis Barreira and Claudia Valls Springer Undergraduate Mathematics Series Complex Analysis and Differential Equations 2012 10.1007/978-1-4471-4008-5_1 Springer-Verlag London 2012

1. Basic Notions

Luis Barreira 1

(1)

Departamento de Matemtica, Instituto Superior Tcnico, Lisboa, Portugal

Luis Barreira (Corresponding author)

Email:

Claudia Valls

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Abstract

In this chapter we introduce the set of complex numbers, as well as some basic notions. In particular, we describe the operations of addition and multiplication, as well as the powers and roots of complex numbers. We also introduce various complex functions that are natural extensions of corresponding functions in the real case, such as the exponential, the cosine, the sine, and the logarithm.

In this chapter we introduce the set of complex numbers, as well as some basic notions. In particular, we describe the operations of addition and multiplication, as well as the powers and roots of complex numbers. We also introduce various complex functions that are natural extensions of corresponding functions in the real case, such as the exponential, the cosine, the sine, and the logarithm.

1.1 Complex Numbers

We first introduce the set of complex numbers as the set of pairs of real numbers equipped with operations of addition and multiplication.

Definition 1.1

The set of complex numbers is the set 2 of pairs of real numbers equipped with the operations

(1.1)

and

(1.2)

for each ( a , b ),( c , d )2.

One can easily verify that the operations of addition and multiplication in () are commutative, that is,

and

for every ( a , b ),( c , d )2.

Example 1.2

For example, we have

and

For simplicity of notation, we always write

thus identifying the pair ( a ,0)2 with the real number a (see Figure ). We define the imaginary unit by

(see Figure ).

Figure 1.1

Real number a and imaginary unit i

Proposition 1.3

We have i 2=1 and a + ib =( a , b ) for every a , b .

Proof

Indeed,

and

which yields the desired statement.

We thus have

Now we introduce some basic notions.

Definition 1.4

Given z = a + ib , the real number a is called the real part of z and the real number b is called the imaginary part of z (see Figure ). We also write

Figure 1.2

Real part and imaginary part

Example 1.5

If z =2+ i 3, then and .

Two complex numbers z 1, z 2 are equal if and only if

Definition 1.6

Given z in the form

(1.3)

with r 0 and , the number r is called the modulus of z and the number is called an argument of z (see Figure ). We also write

Figure 1.3

Modulus, argument and polar form

We emphasize that the number in () holds, then

One can easily establish the following result.

Proposition 1.7

If z = a + ib , then

(1.4)

and

(1.5)

where tan1 is the inverse of the tangent with values in the interval (/2,/2).

It follows from () that

(1.6)

Example 1.8

If , then

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