Lecture # 4 (Complex Numbers).pdf deep learning

Complex data

1. Complex
Variables &
Transforms

Instructor: Dr. Naila Amir
MATH- 232

2. Complexnumbers
▪Complex Numbers and Their Properties
▪Complex Plane
▪Polar Form of Complex Numbers
▪Powers of Complex Numbers
▪Roots of Complex Numbers
▪Sets of Points in the Complex Plane
▪Applications

3. We can find integer powers of a complex number � by using results of multiplication
and division in polar form. If �=??????(cos??????+?????? sin??????), then
� �
2
=��=??????
2
cos(??????+??????) +??????sin(??????+??????) =??????
2
cos2??????+??????sin2??????,
� �
3
=�
2
�=??????
3
cos(2??????+??????) +??????sin(2??????+??????) =??????
3
cos3??????+??????sin3??????,
� �
4
=�
3
�=??????
4
cos(3??????+??????) +??????sin(3??????+??????) =??????
4
cos4??????+??????sin4??????,
and so on. Also,
1
�
=
1
??????
cos(0−??????)+??????sin(0−??????)=??????
−1
cos(−??????)+??????sin(−??????).
Similarly,
1
�
2
=�
−2
=??????
−2
cos(−2??????)+??????sin(−2??????).
Integer Powers of ??????

4. Continuing in the same manner, we obtain a formula for the ??????th power of � for any
integer ?????? as:
�
??????
=??????
??????
cos????????????+??????sin????????????=??????
??????
cis????????????=??????
??????
??????
??????????????????
. ∗
Eq. (∗) is the general form of the De Moivre's theorem which is stated as:
“If �=cos??????+??????sin?????? such that �=??????=1 the for any integer ??????
�
??????
=cos????????????+??????sin????????????=cis????????????=??????
??????????????????
. ”
Note: This theorem is also valid if ?????? is a rational number.
Integer Powers of ??????

5. Determine �
3
if �=
3
2
+??????
1
2
.
Solution:
For the present case we have:
�=1 and Arg�=??????=
�
6
.
Thus,
�
3
=
3
2
+??????
1
2
3
=cos
�
6
+??????sin
�
6
3
=cos
3�
6
+??????sin
3�
6
=cos
�
2
+??????sin
�
2
=??????.
Example

6. De Moivre’sTheorem
▪De Moivre's theorem, namedaftertheFrenchmathematicianAbrahamdeMoivre
is used the find the powers and roots of complex numbers.
▪De Moivre's Theorem states thatthe power of a complex number in polar form is
equal to raising the modulus to the same power and multiplying the argument
by the same power.
▪De Moivre's theorem can be derived from Euler's equation, and is important
because it connects trigonometry to complex numbers.
▪In general, for any complex number�and any integer ??????, the following is true:
If �=??????cos??????+??????sin?????? then �
??????
=??????
??????
cos????????????+??????sin????????????.

7. Given �=1+??????3, find
?????? �
2
???????????? �
5
?????????????????? �
7
.
Solution:
For the present case note that:
�=1+3=2,
and
tan??????=
3
1
⇒??????=
�
3
.
Thus,
�=2cos
�
3
+??????sin
�
3
.
Example

8. Since,
�=2cos
�
3
+??????sin
�
3
,
so
?????? �
2
=4cos
2�
3
+??????sin
2�
3
=4
−1
2
+??????
3
2
=−2+23??????.
???????????? �
5
=2
5
cos
5�
3
+??????sin
5�
3
=32cos
−�
3
+??????sin
−�
3
=16−163??????.
?????????????????? �
7
=2
7
cos
7�
3
+??????sin
7�
3
=128cos
�
3
+??????sin
�
3
=64+633??????.
Example

9. Book: A First Course in Complex Analysis with Applications by Dennis G.
Zill and Patrick D. Shanahan.
Chapter: 1
Exercise: 1.3
Q # 1 – 38.
Practice Questions

10. Applications

11. Complex Numbers
Book: A First Course in Complex Analysis with Applications by
Dennis G. Zill and Patrick D. Shanahan.
•Chapter: 1
•Sections: 1.4, 1.6

12. Applications of De Moivre’s Theorem
▪To express cos???????????? and sin???????????? as finite sums of trigonometric
functions of ??????, where ?????? is a positive integer.
▪To express powers of cos?????? (or sin??????) in a series of cosines (or
sines) of multiples ??????.
▪To find ??????th roots of a complex number.

13. Roots of a Complex Number
Suppose ??????cos??????+??????sin?????? and �cos??????+??????sin?????? are polar forms of the complex
numbers � and ?????? respectively. Then, the equation: ??????
??????
=�, becomes:
�
??????
cos????????????+??????sin????????????=??????cos??????+??????sin?????? (1)
From (1), we can conclude that:
�
??????
=?????? (2)
and
cos????????????+??????sin????????????=cos??????+??????sin?????? (3)
From (2), we define:�=??????
1/??????
to be the unique positive ??????th root of the positive
real number ??????. From (3), the definition of equality of two complex numbers
implies that:
cos????????????=cos?????? and sin????????????=sin??????.
These equalities, in turn, indicate that the arguments ?????? and ?????? are related by the
equation ????????????=??????+2??????�, where ?????? is an integer.

14. Roots of a Complex Number
Thus,
??????=
??????+2??????�
??????
.
As ?????? takes on the successive integer values ??????=0,1,2,… ,??????−1, we obtain ??????
distinct ??????th roots of �; these roots have the same modulus ??????
1/??????
but different
arguments. Thus, the ?????? ??????th roots of a non-zero complex number � are given by:
??????
??????=�
1/??????
=??????
1/??????
cos
??????+2??????�
??????
+??????sin
??????+2??????�
??????
, (4)
where ?????? = 0,1,2,...,?????? − 1. These ?????? values lie on a circle of radius ??????
1/??????
with
center at the origin and constitute the vertices of a regular polygon of ?????? sides. The
value of �
1/??????
obtained by taking the principal value of arg� and ?????? = 0 in (4) is
called the principal ??????th root of ??????=�
1/??????
.

15. Example
Determine the four fourth roots of � =2(1+??????).

16. Example
Note that, since Arg(�)=�/4, so we have:
??????
0≈1.1664+?????? 0.2320,
is the principal fourth root of � =2(1+??????).
As shown in figure, the four roots lie on a circle
centered at the origin of radius ??????=
4
2≈1.19
and are spaced at equal angular intervals of
2�/4=�/2radians, beginning with the root
whose argument is �/16.

17. Example
Determine the cube roots of � = ??????.

18. Example
Note that, since Arg(�)=�/2, so we have:
??????
0≈0.8660 + 0.5??????,
is the principal cubic root of � =??????.
As shown in figure, the three roots lie on a circle
centered at the origin of radius ?????? ≈ 1 and are
spaced at equal angular intervals of 2�/3
radians, beginning with the root whose
argument is �/6.

19. Quadratic Formula
The quadratic formula is perfectly valid when the coefficients �≠0,�, and � of a
quadratic polynomial equation ��
2
+��+�=0 are complex numbers. Although
the formula can be obtained in exactly the same manner as for a quadratic
polynomial ��
2
+��+�=0 where the coefficients �≠0,�, and � are real.
However, we choose to write the result as:
�=
−�+�
2
−4��
1/2
2�
. (∗)
Notice that the numerator of the right-hand side of (∗) looks a little different than
the traditional −�±�
2
−4��. Keep in mind that when �
2
−4��≠0, the
expression�
2
−4��
1/2
represents the set of two square roots of the complex
number�
2
−4��. Thus, (∗) gives two complex solutions.

20. Example
Solve the quadratic equation �
2
+1 − ??????� − 3?????? = 0.
Solution:
Using quadratic formula with �=1,�=1−??????, and �=−3??????, we have:
�=
−1−??????+1−??????
2
−4(−3??????)
1/2
2
=
−1+??????+10??????
1/2
2
.
To compute10??????
1/2
we will use the procedure for finding roots of any complex
numbers. For the present case �=10?????? with ??????=10 and ??????=�/2. We are required
to determine two square roots of 10??????. Here ??????=2 and ??????=0,1. Thus, the two square
roots of 10?????? are given as:
??????
0=10cos
�
4
+??????sin
�
4
=51+??????,
and
??????
1=10cos
5�
4
+??????sin
5�
4
=−51+??????.

21. Example
Thus,
�=
−1+??????+10??????
1/2
2
.
possess two root which are given as:
�
1=
1
2
−1+??????+5+??????5=
5−1
2
+??????
5+1
2
,
and
�
2=
1
2
−1+??????−5−??????5=
−5+1
2
−??????
5−1
2
.

22. Factoring a Quadratic Polynomial
By finding all the roots of a polynomial equation we can factor the polynomial
completely. This statement follows as a corollary to an important theorem that will
be proved later. For the present, note that if �
1 and �
2 are the roots defined by (∗),
i.e.,
�=
−�+�
2
−4��
1/2
2�
. (∗)
Then aquadraticpolynomial��
2
+��+�factorsas:
��
2
+��+�=��−�
1�−�
2. (∗∗)

23. Example
Factorize the quadratic polynomial �
2
+1 − ??????� − 3?????? .
Solution:
Using equation (**), we have
�
2
+1 − ??????� − 3??????=�−�
1�−�
2
=�−
5−1
2
−??????
5+1
2
�+
5+1
2
+??????
5+1
2
.

24. Practice Questions
1.Find the three cube roots of �=−1+??????.
2.Find the squares of all the cube roots of �= − ??????.
3.Find the four fourth roots of �=−23+2??????.
4.Find the squares of all the 5th roots of �=
1
2
+
3
2
??????.
5.Find the six 6th roots of (i) −1 and (ii) 1+??????.

25. Book: A First Course in Complex Analysis with Applications by Dennis G.
Zill and Patrick D. Shanahan.
Chapter: 1
Exercise: 1.4
Q # 1 – 20, 25 – 26.
Exercise: 1.6
Q # 1 –12.
Practice Questions

26. Complex Magic
▪Every complex number can be transformed into polar form. For � = ?????? , we obtain:
??????=??????
????????????/2
.
Thus, in reality ?????? correspond to rotation by
??????
2
radians (90 degrees). This is the reason
imaginary part is always sketched on the �−axis as it is at perpendicular to �−axis.
▪Suppose we have two complex numbers expressed in exponential form �
1=??????
1??????
????????????1
and �
2=??????
2??????
????????????2, then their product is defined as:
�
1�
2=??????
1??????
????????????1??????
2??????
????????????2=??????
1??????
2 × ??????
??????(??????1+??????2)
.
re-scaling
▪Therefore,weconcludethatacomplexproductencodesaninformationoftworeal
physicaloperationsre-scalingandrotation.Wecanthinkonsimilarlinesaboutother
complexoperationse.g.,addition,division,complexconjugation.
rotation

27. Complex Magic
Letusexplorethecomplexdomaininmoredetail.Considerarealquadraticequation
�
2
− 1 = 0.
Weknowthatthesolutionsofthisequationare� = ±1, whicharetwopointsonthe
realline.

28. Complex Magic
▪Now consider the complex equation
�
2
−1=0.
▪How many solutions does this complex equation has?
▪Of course, two: � = ±1, two points on the complex plane. If the answers are same, then
we might believe that both equations are same !!!.
▪It is certainly not true.
▪The beauty of second equation can only be seen if we substitute � = � + ??????� , into the
equation, i.e,
(�+??????� )
2
− 1 = 0
�
2
−�
2
+2??????�� − 1 = 0
�
2
−�
2
− 1 + ?????? 2��=0
which gives rise to two equations:
�
2
−�
2
= 1 and 2�� = 0,
▪The first is an equation of a hyperbola and the other is an equation of � − or � −axis.

29. Complex Magic
Therefore, we can conclude that the two solutions � = ±1, are basically two points on
the intersection of hyperbolas and � − axis.

30. Some important facts about complexnumbers
1.Space of all complex numbers ℂ, is a vector space.
2.
1
�
=
ҧ�
�
2
, where �=�+??????�.
3.�
1+�
2≤�
1+�
2. (triangular inequality)
4.An ??????th degree complex polynomial equation has n complex roots. On the other
hand, an ??????th degree real polynomial equation may or may not have ?????? real roots.
5.If�=�+??????�wedefine??????
�
=??????
�+??????�
=??????
�
cos�+??????sin�.
6.??????
�
is never zero.
7.??????
�
=??????
�
, where �=�+??????�.