Summary Mathematics class notes on topic "Complex analysis and numerical methods"

E-Material
Complex Analysis & Numerical Methods
SUBJECT CODE: CUTM1003 CREDIT: 2+0+1

Module-I
Basic Concept of Complex Numbers

Introduction

Complex Number: The number z which is written in the form z  x  iy , where x, y  R is called a
complex number.

Note: We extend R to C i.e. R  C.

Any real number x can be written in the form of complex number as x  x  0 i .

The numbers x and y are respectively called as the real and imaginary parts of z and are written as:

x  Re  z  ; y  Im  z 

For this reason the complex number x  iy is denoted as the ordered pair  x, y  .

Conjugate and Modulus of a Complex number:

Let z  x  iy be a complex number.

Conjugate: The conjugate (complex conjugate) of z ,

Denoted as z , is defined as z  x  iy

Note: Any real number x is its own conjugate.

Modulus: The modulus or absolute value of z , denoted as z is defined as z   x2  y 2

Some properties of Conjugate and Modulus:

(i) The conjugate of the sum or difference of two complex numbers is the sum of their conjugates i.e. if
z1 , z2  ; then z1  z2  z1  z2 .
(ii) The conjugate of the product of two complex numbers is the product of their conjugates i.e. if
z1 , z2  ; then z1.z2  z1.z2

,(iii) The conjugate of the quotient of two complex numbers (with non-zero denominator) is the

quotient of their conjugates i.e. if z1 , z2 
  z
with z2  0 ; then  z1   1
z
 z2  2

(iv) The product of a complex number and its conjugate is equal to the square of the modulus of the
, then z · z  z
2
complex number i.e. if z
zz zz
(v) If z  x  iy , then x  Re  z   and y  Im  z  
2 2i
(vi) z  z  z is real and z   z  z is imaginary.
1 z
(vii) For z  0;  2
z z
(viii) For any z  ;
Re  z   Re  z   z and Im  z   Im  z   z

(ix) A complex number and its conjugate have the same modulus i.e. z  z .
(x) The modulus of the product and quotient of two complex numbers is equal to the product and
quotient of their moduli respectively. i.e.
z1 z
z1 z2  z1 z2 and  1 ; z2  0 .
z2 z2

Complex Plane & Geometrical Representation of Complex Numbers:

We know how to plot a point  x, y  in the Cartesian plane. Identifying the same point with the
complex number  x  iy  , we get the Complex plane or Argand plane.

I (Imaginary axis)
M P (x, y)

r

 M
y R (Real axis)
O x


N (x, –y)




Consider a complex plane and a point P  x, y  on it which is identified as z  x  iy .

Draw PM  X  axis. Also draw the  to N such that PM  MN . Now the coordinate of N is
 x,  y  . So the complex number associated with N is x  iy  z

,Thus z represents the complex number associated with the reflexion of the point P which denotes the
complex number z. It follows from the right-angled triangle OMP that,

OP 2  OM 2  MP 2 (By Pythagoras theorem)

 x2  y 2

 OP  x2  y 2  z

Triangle Inequality of Complex Numbers z1  z2  z1  z2 ; z1 , z2 


Proof: Now z1  z2   z1  z2  z1  z2
2
 
  z1  z2  z1  z2 
 z1 z1  z1 z2  z2 z1  z2 z2
 z1  z1 z2  z1 z2  z2
2 2



 z1  2 Re  z1 z2   z2
2 2



 z1  2 z1 z2  z2
2 2
 Re  z   z 
 z1  2 z1 z2  z2
2 2
 z2  z2 
  z1  z2 
2




(xi) z1  z2  z1  z2

Proof: Now z1   z1  z2   z2  z1  z2  z2


 z1  z2  z1  z2



Parallelogram Law

Show that z1  z2  z1  z2  2 z1  z2
2 2
 2 2
;  z , z 
1 2



z1  z2  z1  z2
2 2
Proof:


 
  z1  z2  z1  z2   z1  z2  z1  z2  
  z1  z2  z1  z2    z1  z2  z1  z2 

, 
 z1  2Re  z1 z2   z2
2 2
 z
1
2
 z2  2Re  z1 z2 
2


 2 z1  z2
2 2

Some Basic Definitions

Circle: A general circle of radius  and center a is defined as z  a  . It is the set of all z whose

distance z  a from centre a equals  .
y

(But a unit circle is z  1 ).  z
a

x x
O




y




Its interior (“open circular disk”) is given by z  a   , its interior points plus circle itself (closed

circular disk) is given by z  a   , and its exterior by z  a  .




Neighborhood: An open circular disk z  a   is also called a neighbourhood of a. Thus the
neighbourhood of a consists of all points lying inside but on a circle of radius  with centre at a .

Annulus:

1) Closed Annulus: The set  z : 1  z  a   2  including two circles is called a closed annulus.

2) Open Annulus:The set  z : 1  z  a   2  is called an open annulus and it is the set of all z

whose distance z  a from a is greater than 1 but less than  2
Half planes: By the (open) upper-half plane we mean the set of all points z  x  iy s.t. y  0 . Similarly
the condition y  0 defines the lower half plane, x  0 the right-half plane and x  0 the left half plane.

Set: A point set in the complex plane means any sort of collection of finitely many or infinitely many
points.

Limit point: A point z0 is said to be a limit point of set of points S  , if every neighborhoodof z0
contains a point of S other than z0 .