adv math[unit 1]

Advanced Engineering Mathematics

Complex Numbers

COMPLEX NUMBERS

1.1 Complex Numbers. Complex Plane

Equations without real solutions, such as x� =−1 or x� − 10x + 40 = 0, were observed early in history and led to the introduction of complex numbers. By definition, a complex number z is an ordered pair �x, y� of real numbers x and y, written in rectangular form as

z = x + jy (1.1)

where x is the real part of z, y is the imaginary part of z, written

x = Re�z� y = Im�z�

(1.2a) (1.2b)

and j (in theoretical mathematics, i) as the imaginary unit which is equal to

j = √−1 (1.3)

By definition, two complex numbers are equal if and only if their real parts are equal and their imaginary

parts are equal, or if z� =x� + jy� is equal to z� = x� + jy� then

Re�z�� = Re�z�� x� =x�

(1.4a) (1.4b)

and

Im�z�� = Im�z�� y� =y�

(1.4c) (1.4d)

The addition of complex numbers z� and z� is defined as

z� + z� = �x� + x�� + j�y� + y�� (1.5)

or the sum of two complex numbers is the sum of their real parts plus the sum of their imaginary parts

multiplied by the imaginary unit. The addition of complex numbers is similar to vector addition.

The multiplication of complex numbers z� and z� is defined as

z�z� = �x�x� −y�y�� + j�x�y� +x�y�� (1.6)


Complex Numbers

Example 1.1

Let z� = 8 + j3 and z� = 9− j2. Find (a) the real part of z� and z�. (b) the imaginary part of z� and z�. (c) the sum z� +z� (d) the product z�z� Answers:

(a) Re�z�� = 8 and Re�z�� = 9 (b) Im�z�� = 3 and Im�z�� = −2 (c) z� +z� = 17 + j (d) z�z� = 78 + j11

Subtraction is defined as the inverse operation of addition. Thus the difference z� −z� is defined as

z� − z� = �x� − x�� + j�y� −y�� (1.7)

Division is defined as the inverse operation of multiplication. The quotient �� is defined as

z�z�

=x�x� +y�y�x�� +y��+ j x�y� −x�y�x�� +y��

(1.8)

Example 1.2 Let z� = 8 + j3 and z� = 9− j2. Find (a) the difference z� −z�. (b) the quotient

��.

Answers:

(a) z� −z� = −1 + j5 (b)

��= !" + j #$!"

Since the complex number z = x + jy contains an ordered pair �x, y�, it can be plotted in a standard Cartesian coordinate plane. We choose the horizontal axis as the x-axis and is called the real axis and the

vertical axis as the y-axis called the imaginary axis. We choose the same unit length for both axes. A

complex number z = x + jy is represented as a point P�x, y�, shown in Figure 1.1. The coordinate axes in which complex numbers are represented are called the complex plane, or sometimes, the Argand

diagram.


Complex Numbers

Figure 1.1 The Complex Plane

Figure 1.2 shows how the complex number z = 4 − j3 is represented in the complex plane.

Figure 1.2 Example of a point z in complex plane

The complex conjugate z∗ (in some books z') of a complex number z = x + jy is defined by

z∗ = x − jy (1.9)

It is obtained geometrically by reflecting the point z in the real axis. Figure 1.3 demonstrates this.

Figure 1.3 Geometric interpretation of conjugation


Complex Numbers

Drill Problems 1.1

For items 1 to 9, let z� = 2 + j3 and z� = 4− j5 . Showing the details of your work, find (in the rectangular form x + jy):

1. �5z� + 3z�� 2. z�∗z�∗ 3. Re( �

��)

4. Re�z��, *Re�z��+�

5. ��

6. ��∗��∗, (��)

∗

7. �4z� −z�� 8.

��∗��, ��∗

9. ��,��-��

For items 10 to 13, let z = x + jy. Find in the rectangular form: 10. Im�z$�, *Im�z�+$ 11. Re ( ��∗) 12. Im.�1 + j�!z�/ 13. Re ( �

�∗�) 14. Verify the following laws of conjugation:

�z� + z��∗ =z�∗ +z�∗ �z�z��∗ =z�∗z�∗

15. Show that j� =−1, j$ =−j , j# = 1 and �0 =−j, �0� = −1, �01 = j, �02 = 1. From the results of these, evaluate j�$�� and its reciprocal, j-�$��.

1.2 Polar Form of Complex Numbers. Powers and Roots.

The complex plane becomes even more useful and gives further insight into the arithmetic operations for

complex numbers if besides the xy-coordinates (the rectangular form) we also employ the usual polar

coordinates r and θ, defined by

x = r cosθ y = r sin θ

(1.10a) (1.10b)

We see that then z = x + jy takes the so-called polar form

z = r�cosθ + j sin θ� = r∠θ (1.11)


Complex Numbers

The parameter r is called the absolute value or modulus or magnitude of z denoted by |z|. Hence,

|z| = r = ;x� +y� =√zz∗ (1.12)

Geometrically, r is the distance of the point z from the origin. Similarly, |z� −z�| is the distance between z� and z�.

The parameter θ is called the argument or angle of z and is denoted by arg�z�. Thus,

arg�z� = θ = arctan ?@ (1.13)

Geometrically, θ is the directed angle from the positive x-axis to the terminal point z. Here, all angles are measured positive in the counterclockwise sense. See Figure 1.4.

Figure 1.4 Polar coordinates in complex plane

The principle value of the angle θ, denoted as Arg�z� (with a capital A) is a unique value for the angle θ which lies in between –π and π. Thus,

−π D Argz E π (1.14)

for z F 0 . For a given complex number, the other values of arg z are arg z = Argz G 2πn�n =G1, G2, . . . �.


Complex Numbers

Example 1.3

For z� = 1 + j, z� =−1 − j, and z$ = 3 + j3√3, find (a) the modulus r, the principal argument θ and express each in polar form. (b) all the possible arguments. (c) the plot of each one in complex plane. Answers: For z�, (a) r� =√2, Argz� = �#π, z� =√2 (cos �#π + j sin �

#π) = √2∠ �#π;

(b) arg z� = �#π G 2πn �n = G1,G2, . . . � For z�, (a) r� =√2, Argz� =− $I

# , z� = √2(cos (− $#π) + j sin (− $

#π)) = √2∠(− $#π);

(b) arg z� =− $#π G 2πn �n = G1,G2, . . . �

For z$, (a) r$ = 6, Argz$ =I$, z$ = 6 (cos(�$π) + j sin (�$π)) = 6∠ (�$π); (b) arg z$ = �$π G 2πn �n = G1,G2, . . . �

Multiplication and division of complex numbers are easier to perform in polar rather than the rectangular

form. If we have z� =x� + jy� = r�∠θ� and z� =x� + jy� = r�∠θ�, we can prove that

z�z� = r�r��cos�θ� +θ�� + j sin�θ� +θ�� = r�r�∠�θ� +θ�� (1.15)

Thus, |z�z�| = |z�||z�| and arg�z�z�� = arg z� + arg z�. Similarly, for division, z�z�

= r�r��cos�θ� −θ�� + j sin�θ� −θ�� = r�r�

∠�θ� −θ�� (1.16)

Thus, K��K = |��||��|and arg (��) = arg z� − arg z�.


Complex Numbers

Example 1.4

Given z� =−2 + j2 and z� = j3, (a) Find the product z�z� and the quotient �� in rectangular form, without converting to polar form. (b) Find the product z�z� and the quotient �� by converting first to polar form, then back to rectangular form. Answer:

z�z� =−6 − j6 z�z�

=23 + j 23

The integer power of a complex number z, zL, can be found by induction from Eq. 1.15 by

zL = rL�cosnθ + j sin nθ� = rL∠nθ (1.17)

For |z| = r = 1, Eq. 1.17 becomes the De Moivre’s formula

�cosθ + j sin θ�L = cosnθ + j sin nθ (1.18)

Example 1.5 Evaluate the following, expressing the answer in rectangular form:

(a) �3 + j4�$ (b) �1 + j�� Answers: (a) −117 + j44 (b) −64

If z = wL, then there are n complex values of z which will satisfy the equation. Those values, or roots, can be found, from De Moivre’s formula as

√zN = √rN (cos O,�PIL + j sin O,�PIL ) (1.19)

where k = 0, 1, 2, . . . , n − 1. These n values lie on a circle of radius √rN with center at the origin and

constitute the vertices of a regular polygon on n sides. The value of √zN obtained by taking the principal

value of arg z and k = 0 in Eq. 1.19 is called the principal root.


Complex Numbers

Figure 1.5. Plot of nth root of 1, for n = 3, 4 and 5 respectively

Example 1.6 Find all the roots of the following in rectangular form and plot them:

(a) ;−j (b) ;−5 − j12 Answers:

(a) √�� − j √�� , −

√�� + j √��

(b) G�2 − j3�

Drill Problems 1.2 For items 1 to 8, represent the following numbers in polar form. Show the details of your work. 1. 3 − j3 2. j2, −j2 3. −5 4.

��+ j �#π

5. �,0�-0

6. $√�,0�

-√�-0�� $⁄ �

7. - ,0"0$

8. �,0$",0#

For items 9 through 11, find all the roots of the given expression in rectangular form. Plot the roots in the complex plane.

9. √−12

10 ;3 + j41

11. √1S


Complex Numbers

For items 12 and 13, evaluate the following, expressing the final answers in rectangular form.

12. �9 + j9�$ 13. �−2 + j6�� For items 14 and 15, prove the following trigonometric identities using De Moivre’s formula:

14. cos 2θ = cos�θ −sin�θ 15. sin 2θ = 2 cos θ sin θ

1.3 Exponential Form of Complex Numbers. Trigonometric and Hyperbolic Forms.Logarithms and General

Power.

Complex number z = x + jy can be written in an exponential form

z = re0O (1.20)

where r and θ are the magnitude and argument of z as discussed in the previous section. Taking the polar form of z,

z = r�cosθ + j sin θ� = r∠θ (1.11)

we see that

e0O = cos θ + j sin θ (1.21a)

which is called the Euler’s formula. The conjugate of this is

e-0O = cos θ − j sin θ (1.21b)

The exponential form of complex numbers has many interesting properties, which relates exponential,

trigonometric and hyperbolic functions of complex numbers. For example, adding Eqs. 1.21a and 1.21b

gives the exponential equivalent of cosine function, that is

cosθ = e0O +e-0O

2 (1.22a)

and subtracting the two gives the exponential equivalent sine function

sin θ = e0O −e-0O

j2 (1.22b)


Complex Numbers

Hyperbolic cosine and sine functions are defined as follows:

cosh z = e� +e-�2 (1.23a)

sinh z = e� −e-�2 (1.23b)

If we replace z = jθ into Eqs. 1.23a and 1.23b, they become

cosh jθ = e0O +e-0O

2 = cosθ (1.24a)

sinh jθ = e0O +e-0O

2 = j sin θ (1.24b)

Similarly, we can prove the following identities

cos jθ = cosh θ (1.25a)

sin jθ = j sinh θ (1.25b)

Take note that the exponential form of complex numbers is also periodic, following the periodic properties

of cosine and sine functions. Thus, for a complex number e�,

e�G0�I = e� (1.26)

Example 1.7 Evaluate the following expressions, expressing answers in rectangular form

(a) cos�1 + j� (b) sinh�4 − j3� Answers:

(a) 0.8337 − j0.9889 (b) −27.0168− j3.8537


Complex Numbers

The natural logarithm of z = x + jy is denoted by ln z and is defined as the inverse of the exponential function; that is, w = lnz is defined for z F 0 by the relation eV = z. By setting w = u + jv, we find that the logarithm of a complex number z is given as

ln z = ln r + jθ (1.27)

where r = |z| > 0 and θ = arg z.

Recall that the argument of z is determined only up to integer multiples of 2π, it follows that the complex natural logarithm is infinitely many-valued. The value of ln z corresponding to the principal value Argz is denoted by Lnz and is called the principal value of ln z. Thus,

Ln z = ln|z| + jArgz (1.28)

The uniqueness of Argz for given z implies that Lnz is single-valued, that is, a function in the usual sense. Since, the other values of arg z differ by integer multiples of 2π, the other values of ln z are given by

ln z = Lnz G j2πn (1.29)

Example 1.8 Evaluate the following logarithms, expressing the answers in rectangular form: (a) ln 1, Ln1 (b) ln�3 − j4�, Ln�3 − j4� Answers: (a) ln 1 = 0,Gj2πn, Ln1 = 0 (b) ln�3 − j4� = 1.609 − j0.927 G j2πn, Ln�3 − j4� = 1.609 − j0.927

The general powers of a complex number z = x + jy are defined by the formula

z[ =e[ \L � (1.30)

where c is also a complex number, and z F 0. Since ln z is infinitely-valued, z[ , will, in general, be multivalued. The particular value

z[ =e[ ]L � (1.31)

is called the principal value of z[.


Complex Numbers

Example 1.9 Evaluate the following, expressing the answers in rectangular form:

(a) j0 (b) �1 + j��-0� Answers:

(a) e-�I �⁄ �∓�IL

(b) 2eI #⁄ G�IL _sin (�� ln 2) + j cos (�� ln 2)`

Drill Problems 1.3

For numbers 1 through 4, find the principal value of ln z, rectangular form, when z equals: 1. −10 2. 2 + j2 3. 2 − j2 4. – je For numbers 5 through 8, evaluate the following, expressing the answers in rectangular form.

5. ln e 6. ln−e-0 7. ln�4 + j3� 8. ln e0$ For numbers 9 through 12, find the principal value in rectangular form:

9. j0�, j20 10. 4�$,0� 11. �1 − j��,0� 12. �−1��-0�� For numbers 13 through 15, solve for z, in rectangular form: 13. ln z = (2 − j ��) π 14. ln z = 0.3 + j0.7 15. ln z = e − jπ