Functions of Complex Variables



Functions of Complex Variables

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| |This unit is mainly devoted in presenting basic concepts on Complex Numbers, Complex Analytic Functions, the Cauchy-Riemann | |

| |Equations, Laplace’s Equations, Elementary Complex Functions (Exponential Functions, Ttrigonometric Functions and Hyperbolic | |

| |Functions), Line Integral in the Complex Plane, Cauchy’s Integral Theorems, Derivatives of Analytic Functions, Power Sseries, Taylor| |

| |Series, Laurent Series, Residue Integration and Evaluation of Real Integrals. | |

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4.1 Definition of Complex Numbers

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| |The concept of complex number basically arises from the need of solving equations that has no real solutions. Though the Italian | |

| |mathematician GIROLAMO CARDANO used the idea of complex numbers for soving cubic equation the term “complex numbers” was introduced | |

| |by the German mathematician CARL FRIEDRICH GAUSS. | |

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|Definition 4.1 A complex number z is an ordered pair (x, y) of real numbers |

|x and y, written z = (x, y), x is called the real part and y the imaginary |

|part of z, usually the real and imaginary parts of the complex number |

|z = (x, y) are denoted by |

|x = Re z and y = Im z. |

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|Definition 4.2 Two complex numbers are equal if and only if their |

|corresponding real and imaginary parts are equal. |

Example 4.1 Find the values of ( and ( for which the complex numbers[pic].

Solution By definition 4.2

[pic] ( [pic] ( [pic].

Therefore, [pic].

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|Definition 4.3 The complex number (0,1) usually denoted by i = (0,1) is |

|called imaginary unit |

4.1.1 Addition and Multiplication on Complex Numbers

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|Definition 4.4 For any two complex numbers [pic] and [pic] |

|i) [pic] ( [pic] = [pic] |

|ii) [pic] |

Note that: Any real number x can be written as x = (x, 0) and hence the set of complex numbers

extend the reals.

Example 4.2 Let[pic],[pic]( (. Then from definition 4.4 we get:

[pic] +[pic] = [pic] and [pic][pic] = [pic]

Furthermore; for any real numbers x and y,

i y = [pic](y, 0) = (0, y) and (x, y) = (x, 0) + (0, y) = x + i y.

Conequentely; for any real numbers x and y,

i y = (0, y) and (x, y) = x + i y.

Note that: 1. For any non-zero real number y, z = i y is called pure imaginary number.

2. Any point on the x-axis has coordinates of the form (x, 0) that corresponds to the

complex number x = x + 0 i, due to this reason the x-axis is called the real axis.

3. Any point on the y-axis has coordinates of the form (0, y) that corresponds to the

complex number i y = 0 + i y, and hence it is called the imaginary axis.

4.1.2 Properties of Addition and Multiplication

Let [pic], [pic]and [pic] be complex numbers. Then

i) [pic]+[pic] = [pic]+ [pic]and [pic][pic] = [pic][pic]

ii) ([pic]+[pic]) + [pic] = [pic]+ ([pic]+[pic]) and ([pic][pic])[pic]=[pic]([pic][pic])

iii) [pic]([pic] +[pic]) = [pic][pic] +[pic][pic]

iv) 0 +[pic]= [pic], [pic]+ (( [pic]) = 0 and[pic]= [pic]

Furthermore; for any non-zero complex number z = x + i y, there is a complex number [pic] such that [pic].

The complex number [pic]is usually denoted [pic].

Consequentely;

[pic]= [pic] = [pic]= [pic].

Therore, any non-zero complex number z = x + i y has a unique multiplicative inverse given by:

[pic] = [pic].

The set of complex numbers form a field. However, it is not possible to define an order relation on the set of complex numbers. Since the expressions like z > 0, [pic]< [pic]etc are meaningless unless these complex numbers are reals.

4.1.3 Complex Plane

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| |The concept of expressing a complex number (x, y) as a point in the coordinate plane was first introduced by Jean Robert Argand | |

| |(1768-1822), a swiss bookkeeper. The plane formed by a one to one correspondence of complex numbers and points on the coordinate | |

| |plane is called the Argand diagram, or the complex plane or the z-plane. | |

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In the Argand diagram the x-axis is the real axis and the y-axis is called the imaginary axis

|[pic] |In a complex plane any complex number |

| |z = x + i y is represented as the point z with co-ordinate x and |

| |ordinate y, and we say the point z in the complex plane. |

The sum of two complex numbers can be geometrically interpreted as the sum of two position vectors in the Argand diagram.

|[pic] |

4.1.4 Complex Conjugate

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|Definition 4.5 Let z = x + i y be a complex number. Then the complex conjugate |

|of z (or simply the conjugate of z) denoted [pic] is defined by |

|[pic]= x ( i y |

For any complex number z = x + i y in the complex plane, the complex conjugate of z,[pic]= x ( i y

is obtained by reflecting z in the real axis.

Example 4.3 Let z = x + i y be any complex number. Then verify that

i) [pic] ii) Re z = [pic](z + [pic]) iii) Im z = [pic](z ( [pic])

Solutions Using properties of addition and multiplication on complex numbers and definition 4.5 we get:

i) [pic]= [pic]= [pic]= [pic].

Therefore, [pic]= [pic].

ii) [pic](z + [pic]) = [pic][pic] = x = Re z .

Therefore, Re z = [pic](z +[pic]).

iii) [pic][pic]= [pic][pic]= y = Im z.

Therefore, z = [pic](z ( [pic]).

Example 4.4 Let [pic]and [pic]be two complex numbers. Show that:

i) [pic] = [pic] ( [pic] ii) [pic] =[pic][pic] iii) [pic], provided that[pic]( 0.

Solutions Let [pic]and[pic]. Then From the properties of addition and

multiplication on complex numbers and definition 4.5 we get:

i) [pic] = [pic] = [pic]

= [pic]=[pic]=[pic] ( [pic]

Therefore, [pic] = [pic] ( [pic] for any two complex numbers[pic]and[pic].

ii) [pic] =[pic] = [pic]

= [pic]= [pic]=[pic][pic].

Therefore, [pic] =[pic][pic]for any two complex numbers[pic]and[pic].

iii) [pic]= [pic]= [pic]= [pic]= [pic] = [pic].

Therefore, [pic]= [pic], provided that[pic]( 0.

4.1.5 Polar Form of Complex Numbers

The Cartesian coordinates x and y can be transformed into polar coordinates r and ( by

x = r cos ( and y = r sin (

For any complex number z = x + i y the form

z = r (cos ( + i sin ()

is called the polar form of z, where r is the absolute value or modulus of z. The modulus of z is usually denoted and defined by

[pic] = r = [pic] = [pic]

while ( is called the argument of z and is denoted and defined by

arg z = ( = [pic] , up to multiples of 2(.

The value of ( that lies in the interval ( ( < ( ( ( is called the principal value of the argument of z and denoted by Arg z.

Note that: the value of (, measured in radian, depends on the quadrant in which the complex

number z belongs.

Example 4.5 Write z = (1 + i in polar form.

Solution To write z in polar form first we need to find [pic]and Arg z.

[pic]= ((1 + i) ((1 ( i) = 2 and hence [pic] = [pic]

and ( = arg z = [pic]= [pic] where n ( Z, but z lies in the second quadrant ,

hence, Arg z = [pic].

Therefore, z = [pic][pic].

Example 4.6 Write z = (1 ( i in polar form.

Solution To write z in polar form first we need to find[pic]and Arg z.

[pic]= ((1 ( i) ((1 + i) = 2 and hence [pic] = [pic]

and ( = arg z =[pic]= [pic] where n ( Z, but z lies in the third quadrant ,

hence, Arg z = [pic].

Therefore, z = [pic][pic].

4.1.6 Important Inequalities

For any two complex numbers[pic] and[pic]

[pic]([pic]+[pic] (Triangle Inequality)

To show that this holds true, let [pic]= [pic] and [pic]=[pic].

Then [pic]= [pic]

= [pic]+ [pic] + [pic]

( [pic]+ [pic] + [pic]( [pic]

Therefore, [pic]([pic]+[pic].

Furthermore; for any finite number of complex numbers [pic],[pic], . . . , [pic]

[pic] ( [pic] (Generalized triangle inequality)

Verify! (Hint: use the principle of Mathematical induction on n)

Example 4.7 Let [pic]= [pic] and [pic]= [pic]. Find [pic]and [pic]+[pic].

Solution [pic]= [pic]= [pic]=[pic],

[pic]=[pic]=[pic]= [pic]

and [pic]=[pic]=[pic] =[pic].

Therefore, [pic]( [pic] +[pic].

4.1.7 Multiplication and Division in Polar Form

Let [pic]= [pic]and [pic]= [pic].

Multiplication

[pic]= [pic]

= [pic]

Therefore,[pic] = [pic][pic]and arg ([pic]) = arg ([pic]) + arg ([pic]) up to multiplies of 2(.

Division

The quotient [pic]is the number z = [pic]satisfying z[pic]= [pic].

Thus arg (z[pic]) = arg z + arg [pic]= arg [pic]and [pic] = [pic][pic]= [pic].

Hence, [pic]= [pic] and arg ([pic]) = arg ([pic]) ( arg ([pic]) up to multiplies of 2(.

Therefore, [pic]= [pic].

Example 4.8 Let [pic]= [pic]and [pic]= [pic]. Express [pic][pic] and [pic]in polar forms.

Solution [pic]= [pic]= 2 and [pic]= [pic]= 3

and arg ([pic]) = [pic] = [pic] where n ( Z.

But [pic]lies in the [pic]quadrant, hence Arg [pic]= [pic] arg [pic]= [pic], where n ( Z. But [pic]

lies in the positive imaginary axis, hence Arg [pic]= [pic]. Thus[pic]= [pic]and [pic]= [pic].

Therefore, [pic][pic] = 6 [pic] and [pic]= [pic].

4.1.8 Integer powers of Complex Numbers

For any non-zero complex number[pic]

[pic]for any n ( Z.

In particular if [pic] = 1, then we get the De Moivre formula

[pic]for any n ( Z.

Example 4.9 Use the De moivre formula to show that for any angle (

[pic]and [pic]

Solution If n = 2, then

[pic]

and from the De Moivre formula we get:

[pic]

Therefore, [pic]and [pic]

4.1.9 Roots of Complex Numbers

Suppose Z is a non-zero complex number. Now we need to solve [pic], where n ( N and n ( 1.

Note that: Each values of ( is called an [pic]root of z, and we write

[pic]

Let z = [pic] and ( = [pic].

Then [pic]( [pic]= [pic]

( [pic], cos ( = cos n ( and sin ( = sin n ( .

( [pic], [pic], where k ( Z.

Note that: For any k ( Z, there exist integers m and h such that

k = m n + h, where h ((0, 1, 2, 3, . . . , n ( 1(

Let [pic]. Then [pic]= [pic]

and [pic]= [pic]

Therefore, [pic], where, k = 0, 1, 2, 3, . . . , n ( 1.

Note that: These n values lie on a circle of radius [pic]with center at the origin and constitute the

vertices of a regular n-gon.

The value of [pic]obtained by taking the principal value of arg z is called the principal value of

( = [pic].

Example 4.1.10 [pic]root of unity

Solve the equation [pic]= 1.

Solution Now [pic]=[pic], k = 0, 1, 2, 3, . . . , n ( 1.

If ( denotes the value corresponding to k = 1, then the n values of [pic] can be written as

1,[pic],[pic], . . .,[pic]

Hence let ( = [pic].

Therefore, 1,[pic],[pic], . . .,[pic]are the [pic]roots of unity.

Example 4.1.11 Solve the equation[pic]= 1.

Solution Now [pic]= [pic], where k = 0, 1, 2, 3.

Then for k = 1 we get ( =[pic].

Therefore, 1,[pic],(1 and ([pic] are the [pic]roots of unity.

Note that: The n values of[pic] are:

[pic],[pic],[pic], . . . ,[pic]

where [pic]= [pic] and [pic]is real.

Note that: For any complex number[pic],

[pic]= [pic] [pic] ( [pic],

where [pic].

= [pic].

Therefore, [pic]= ([pic], where [pic].

Exercise 4.1

1. Write in the form x ( i y, where [pic]= 4 ( 5 i and [pic] = 2 + 3 i

i) [pic] ii) [pic] iii) [pic]

2. Find the real and the imaginary parts of i) ( iii) in exercise 1.

3. Let [pic]and[pic]be complex numbers, if [pic][pic]= 0, then show that either[pic]= 0 or[pic]= 0.

4. Compute [pic]

5. Represent [pic]in polar form.

6. Determine the principal value of the argument of

i) [pic] ii) [pic]

7. Represent each of the following in the form [pic]

i) [pic] ii) [pic]

8. Solve the equation

i) [pic] ii) [pic]

9. For any two complex numbers [pic]and [pic] show that

[pic] + [pic]= [pic] (Parallelogram equality)

4.2 Curves and Regions in the Complex Plane

4.2.1 Circles and Disks

The distance between two points z and [pic] in the complex plane is denoted by[pic]. Hence a circle C of radius[pic]and center [pic]can be given by

[pic] = [pic]

In particular the unit circle with center at the origin is given by[pic] = 1

Furthermore;

i) [pic][pic] represents the exterior of the circle C.

iii) [pic] 9 is the exterior of the circle of radius 9 centered at ( 2 + [pic].

4.2.2 Half plane

i) (open) upper half - plane = [pic]

ii) (open) lower half-plane = [pic]

iii) (open) right half plane = [pic]

iv) (open) left -half plane = [pic]

4.2.2.1 Concepts Related to Sets in the Complex Plane

Now we need to define some important terms.

i) Neighborhoods

A delta, δ neighborhood of a point [pic]is the set of all points z such that [pic]< δ where δ is any given positive number. (a deleted δ-neighborhood of [pic]is a neighborhood of [pic]in which the point [pic]is omitted i.e. 0 ................
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