The document discusses the fundamentals of complex numbers, including definitions, properties, and various mathematical operations such as addition, multiplication, and functions of complex variables. It describes analytic functions, harmonic functions, and methods for finding these functions, as well as the concepts of mapping and conformal mapping in complex analysis. Additionally, it touches upon bilinear transformations and their properties, emphasizing their significance in preserving various mathematical relationships.

2.  Complex Number: A number 𝑧 = 𝑎 + 𝑖𝑏 where 𝑖 = −1 , a
and b are any two real numbers, is called complex number.
 Complex Variable: A variable 𝑧 = 𝑥 + 𝑖𝑦 where 𝑖 = −1 ,
x and y are any two real variables, is called complex
number.
 Complex conjugate: Conjugate of 𝑧 = 𝑥 + 𝑖𝑦 is 𝑧 = 𝑥 −
𝑖𝑦
 Modulus (or) absolute value of a complex: The modulus of
a complex number 𝑧 = 𝑥 + 𝑖𝑦 is given by 𝑧 = 𝑥2 + 𝑦2

3.  Argument (or) amplitude
 The argument of z is the angle 𝜃 between the positive real axis and the
line joining the point to the origin
 Polar form of a complex variable: The polar form of a complexvariable
𝑧 = 𝑥 + 𝑖𝑦 is given by
 𝑧 = 𝑟 𝑐𝑜𝑠𝜃 + 𝑖 𝑟𝑠𝑖𝑛𝜃 , 𝑤ℎ𝑒𝑟𝑒 𝑥 = 𝑟 𝑐𝑜𝑠𝜃, 𝑦 = 𝑟𝑠𝑖𝑛𝜃 , 𝜃 is the angle
between the positive real axis and the line joining the point to the origin
and r is the distance of he point from the origin.

4. Euler’s formula: 𝒆𝒊𝜽
= 𝒄𝒐𝒔𝜽 + 𝒊 𝒔𝒊𝒏𝜽
Modules and amplitude: Modulus 𝑧 =
𝑥2 + 𝑦2 = 𝑟2(cos2 𝜃 + sin2 𝜃) = 𝑟
Amplitude
𝑦
𝑥
=
𝑟𝑠𝑖𝑛𝜃
𝑟𝑐𝑜𝑠𝜃
= 𝑡𝑎𝑛𝜃 ⇒ 𝜃 =
tan−1 𝑦
𝑥

5.  Argand Plane
 The Plane which is convenient way to represent any
imaginary number graphically is known as the Argand
Plane. Let z = x + iy. Then Re (z) = x and Im (z) = y

6.  Note: 𝑟𝑐𝑖𝑠𝜃 = 𝑟(𝑐𝑜𝑠𝜃 + 𝑖𝑠𝑖𝑛𝜃)
 Note
 1. 𝑟2
= 𝑧 2
= 𝑥2
+ 𝑦2
= (𝑥 + 𝑖𝑦)(𝑥 − 𝑖𝑦) = 𝑧 𝑧
 2. 𝑧1 + 𝑧2 ≤ 𝑧1 + |𝑧2|
 3. 𝑧1 − 𝑧2 ≥ 𝑧1 − |𝑧2|
 4. 𝑧1 𝑧2 = 𝑧1 𝑧2
 5.
𝑧1
𝑧2
=
𝑧1
𝑧2
 6.(𝑧1 ± 𝑧2) = 𝑧1 ± 𝑧2
 7. 𝑧1 𝑧2 = 𝑧1 𝑧2
 8. 𝑧 𝑛 = 𝑧 𝑛
 9. 𝑧−1
=
𝑧
𝑧 2 , 𝑧 ≠ 0

7.  Function of complex variable
 If for each value of the complex variable 𝑧 =
𝑥 + 𝑖𝑦 in a region R defined on complex plane
there exist corresponding one or more values of
𝑤 = 𝑓(𝑧) = 𝑢(𝑥, 𝑦) + 𝑖𝑣(𝑥, 𝑦), then 𝑤 = 𝑓(𝑧) is
called the function of the complex variable z.

8.  Analytic function (Regular function /
Holomorphic function)
 A single valued function f (z) is said to be analytic
in a region R of complex plane if f (z) has
derivative at each point of R.
 Examples: Any polynomial function (real or
complex) is analytic.
 The exponential function is analytic.
 The trigonometric functions are analytic.
 The logarithmic functions are analytic.

9.  Necessary and Sufficient condition for a
complex function 𝒇(𝒛) = 𝒖(𝒙, 𝒚) + 𝒊 𝒗(𝒙, 𝒚) is
analytic.
 01. 𝑓(𝑧) is continuous defined on the given
region in complex plane
 02. 𝑢 𝑥, 𝑢 𝑦, 𝑣 𝑥 𝑎𝑛𝑑 𝑣 𝑦 are exists
 03. Satisfies the Cauchy-Riemann
(C-R) equations
𝒊. 𝒆 𝒖 𝒙 = 𝒗 𝒚 𝒂𝒏𝒅 𝒖 𝒚 = −𝒗 𝒙

10.  Note:
 1. The operator 𝛻2
=
𝜕2
𝜕𝑥2 +
𝜕2
𝜕𝑦2 is called Laplace
operator
 2. The equation 𝛁 𝟐 𝒇 =
𝝏 𝟐 𝒇
𝝏𝒙 𝟐 +
𝝏 𝟐 𝒇
𝝏𝒚 𝟐 = 𝟎 is called
Laplace equation

11.  Harmonic function (Potential function)
 A real valued function with two variables x and y which is
continuous and possess 2nd order partial derivatives such
that satisfies Laplace equation is called Harmonic function.
 Harmonic conjugate
 If 𝑢(𝑥, 𝑦) and 𝑣(𝑥, 𝑦) are harmonic and 𝑤 = 𝑓(𝑧) =
𝑢(𝑥, 𝑦) + 𝑖 𝑣(𝑥, 𝑦) is analytic then, u is called harmonic
conjugate of v and v is the harmonic conjugate of u.
 Note
 1. u is harmonic if 𝛻2
𝑢 = 𝑢 𝑥𝑥 + 𝑢 𝑦𝑦 = 0 , v is harmonic if
𝛻2
𝑣 = 𝑣 𝑥𝑥 + 𝑣 𝑦𝑦 = 0
 2. u is harmonic then u can be a real part of an analytic
function

12.  Milne-Thomas method
 Case(i). If the real part u of the analytic function f(z) is known
 Step (1) find 𝑢 𝑥 𝑎𝑛𝑑 𝑢 𝑦
 Step (2) find 𝑢 𝑥 𝑎𝑛𝑑 𝑢 𝑦 at (𝑧, 0)
 Step (3) 𝑓(𝑧) = 𝑢 𝑥 𝑑𝑧 − 𝑖 𝑢 𝑦 𝑑𝑧

 Case(ii). If the imaginary part v of the analytic function f(z) is known
 Steps (1) find 𝑣 𝑥 𝑎𝑛𝑑 𝑣 𝑦
 Step (2) find 𝑣 𝑥 𝑎𝑛𝑑 𝑣 𝑦 at (𝑧, 0)
 Step (3) 𝑓(𝑧) = 𝑣 𝑦 𝑑𝑧 + 𝑖 𝑣 𝑥 𝑑𝑧
 Note : Supposed to find the analytic function whose combined value of u and v is given
 [(i.e) the values like 3𝑢 + 2𝑣 , 𝑢 − 𝑣, 𝑢 + 𝑣 etc]
 Step (1) find partial differentiation of given combined value with respect to x
 Step (2) find partial differentiation of given combined value with respect to y
 Step (3) Using C-R equation 𝑢 𝑥 = 𝑣 𝑦 & 𝑢 𝑦 = −𝑣 𝑥 & convert 𝑢 𝑦 𝑎𝑛𝑑 𝑣 𝑦 𝑏𝑦 𝑣 𝑥 𝑎𝑛𝑑 𝑢 𝑥 in the equation received from step (2)
 Step (4) Solve for 𝑢 𝑥 𝑎𝑛𝑑 𝑣 𝑥
 Step (5) 𝑤. 𝑘. 𝑡, 𝑓′
𝑧 = 𝑢 𝑥 + 𝑖𝑣 𝑥 and find 𝑓′
𝑧 = 𝑢 𝑥 + 𝑖𝑣 𝑥 at (𝑧, 0) and integrate
 We will get f(z)

13.  Case(i). If the real part u of the analytic function f(z) is known
 Steps (1) find 𝑢 𝑥 𝑎𝑛𝑑 𝑢 𝑦
 Step (2) Since f(z) is analytic , 𝑢 𝑥 = 𝑣 𝑦 𝑎𝑛𝑑 𝑢 𝑦 = −𝑣 𝑥 and by exact differential
 𝑑𝑣 = 𝑣 𝑥 𝑑𝑥 + 𝑣 𝑦 𝑑𝑦 = −𝑢 𝑦 𝑑𝑥 + 𝑢 𝑥 𝑑𝑦
 Step (3) 𝑣 = 𝑑𝑣 = − 𝑢 𝑦 𝑑𝑥 + 𝑢 𝑥 𝑑𝑦 = − 𝑢 𝑦 𝑑𝑥 + (𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑢 𝑥 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑔 𝑥) 𝑑𝑦
 Case(ii). If the imaginary part v of the analytic function f(z) is known
 Steps (1) find 𝑣 𝑥 𝑎𝑛𝑑 𝑣 𝑦
 Step (2) Since f(z) is analytic , 𝑢 𝑥 = 𝑣 𝑦 𝑎𝑛𝑑 𝑢 𝑦 = −𝑣 𝑥 and by exact differential
 𝑑𝑢 = 𝑢 𝑥 𝑑𝑥 + 𝑢 𝑦 𝑑𝑦 = 𝑣 𝑦 𝑑𝑥 − 𝑣 𝑥 𝑑𝑦
 Step (3) 𝑢 = 𝑑𝑢 = 𝑣 𝑦 𝑑𝑥 − 𝑣 𝑥 𝑑𝑦 = 𝑣 𝑦 𝑑𝑥 + (𝑡𝑒𝑟𝑚𝑠 𝑜𝑓 𝑣 𝑥 𝑛𝑜𝑡 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝑥 )𝑑𝑦

14.  Shortcut method
 Case(i). If the real part u of the analytic function
f(z) is known
 Then 𝑓 𝑧 = 2𝑢
𝑧
2
,
𝑧
2𝑖
− 𝑢 0,0 +
𝑖𝑐 , 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
 Case(ii). If the imaginary part v of the analytic
function f(z) is known
 Then 𝑓 𝑧 = 2𝑖𝑣
𝑧
2
,
𝑧
2𝑖
− 𝑖𝑣 0,0 +
𝑐 , 𝑤ℎ𝑒𝑟𝑒 𝑐 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

15.  Mapping
 Mapping is a mathematical technique used to convert (or map) one mathematical
problem and its solution into another. It involves the study of complex variables.
 Let a complex variable function z = x +iy define in z- plane have to convert (or map) in
another complex variable function f(z) =w = u +iv define in w- plane. This process is
called as Mapping.
 Conformal Mapping
 Conformal Mapping is a mathematical technique used to convert (or map) one
mathematical problem and its solution into another preserving both angles and shape of
infinitesimal small figures but not necessarily their size.
 The process of Mapping in which a complex variable function z = x + iy define in z-
plane
 mapped to another complex variable function f(z) = w = u +iv define in w- plane
preserving the angles between the curves both in magnitude and sense is called as
conformal mapping.
 The necessary condition for conformal mapping— if w = f(z) represents a conformal
mapping of a domain D in the z −plane into a domain D of the w −plane then f(z) is an
analytic function in domain D.

16.  A mapping 𝒘 = 𝒇(𝒛) is said to be conformal
at 𝒛 = 𝒛 𝟎 if 𝒇’(𝒛 𝟎) ≠ 𝟎
 𝒊. 𝒆
𝒅𝒘
𝒅𝒛
≠ 𝟎 𝒂𝒕 𝒛 = 𝒛 𝟎

17.  The point at which the mapping is not conformal, is called
critical point
 𝑖. 𝑒 𝑖𝑓 𝑓′
𝑧0 = 0 , 𝑧 = 𝑧0 is called critical point.
 𝑖. 𝑒
𝑑𝑤
𝑑𝑧 𝑎𝑡 𝑧=𝑧0
= 0 , then z0 is the critical point of the
transformation 𝑤 = 𝑓 𝑧 .
 If the transformation 𝑤 = 𝑓(𝑧) is conformal at a point
𝑧 = 𝑧0 then the inverse transformation
 𝑍 = 𝑓−1(𝑤) is conformal at the same point 𝑧 = 𝑧0
 So the critical points of the transformation w = f(z) is
given by
𝒅𝒘
𝒅𝒛
= 𝟎 𝒂𝒏𝒅
𝒅𝒛
𝒅𝒘
= 𝟎

18.  Isogonal
 The process of Mapping in which a complex
variable function z = x + iy define in z- plane
mapped to another complex variable function
f(z) = w = u +iv define in w- plane preserving
the angles between the curves in magnitude
but not in sense is called as Isogonal mapping

19.  Consider =
𝑎𝑧+𝑏
𝑐𝑧+𝑑
,𝑎𝑑 − 𝑏𝑐 ≠ 0, where z = x + iy
is a complex variable in z-plane and w = u + iv
is a complex variable in the w-plane, also a, b,
c, d are complex numbers and c and d cannot
both zero simultaneously . This transformation
is called bilinear transformation

20.  Note:
 1. The necessary condition for bilinear
transformation is 𝑎𝑑 − 𝑏𝑐 ≠ 0
 2. The inverse transformation 𝑧 =
𝑑𝑤−𝑏
𝑎−𝑤𝑐
is also
bilinear
 3. 𝑎𝑑 − 𝑏𝑐 is called the determinant of the bilinear
transformation 𝑤 =
𝑎𝑧+𝑏
𝑐𝑧+𝑑
 If 𝑎𝑑 − 𝑏𝑐 = 1 then the transformation is said to
be normalized.
 4. Bilinear transformation is 1-1 and onto (bijective)
 5. Every bilinear transformation maps circle or
straight line onto a circle or straight line

21.  The fixed point of the transformation 𝑤 =
𝑎𝑧+𝑏
𝑐𝑧+𝑑
is obtained from solving
 𝑧 =
𝑎𝑧+𝑏
𝑐𝑧+𝑑
𝑜𝑟 𝑐𝑧2 + 𝑑 − 𝑎 𝑧 − 𝑏 = 0

22. Cross ratio
 The cross ratio of four points 𝑧1, 𝑧2, 𝑧3, 𝑧4 is 𝑧1, 𝑧2, 𝑧3, 𝑧4 =
𝑧1−𝑧2 𝑧3−𝑧4
𝑧2−𝑧3 𝑧4−𝑧1
Note
 1. Bilinear transformation preserves cross ratio.
 If 𝑧1, 𝑧2, 𝑧3, 𝑧4 are points in z-plane and 𝑤1, 𝑤2, 𝑤3, 𝑤4 are
their corresponding images on w-plane under a bilinear
transformation their cross ratios are equal
 𝑖. 𝑒 𝑧1, 𝑧2, 𝑧3, 𝑧4 = (𝑤1, 𝑤2, 𝑤3, 𝑤4)
 2. When three points 𝑧1, 𝑧2, 𝑧3 of z-plane and their
corresponding images 𝑤1, 𝑤2, 𝑤3 on w-plane are given then
the associated bilinear transformation can be found by using
𝑧 −𝑧1 𝑧2−𝑧3
𝑧1−𝑧2 𝑧3−𝑧
=
𝑤−𝑤1 𝑤2−𝑤3
𝑤1−𝑤2 𝑤3−𝑤