This document provides an overview of complex analysis, including: 1) Limits and their uniqueness in complex analysis, such as the limit of a function f(z) as z approaches z0. 2) The definition of a continuous function in complex analysis as one where the limit exists at each point in the domain and equals the function value. 3) Analytic functions, which are differentiable in some neighborhood of each point in their domain.

1. Complex Analysis
( 15UMTC61)
-K.Anitha M.Sc., M.Phil.,
-A.Sujatha M.Sc., M.Phil.,
Department of Mathematics ( SF )
V.V.Vanniaperumal College for Women,
Virudhunagar.

2. Limits
• Let f be a function defined at all points z in some neighbourhood of 𝑧0
except possibly for the point 𝑧0itself.
• The limit of f(z) as z approaches 𝑧0is a number 𝑤0,
• ie, lim
𝑧→𝑧0
𝑓 𝑧 = 𝑤0

3. Uniqueness of limit
• When limit of a function f ( z ) exists at a point 𝑧0 , it must be unique.
Proof:
Suppose f ( z ) has two different limits 𝑤0 and 𝑤1
Then for a positive number 𝜀 , there are + ve number’s 𝛿0 and 𝛿1 such that
𝑓 𝑧 − 𝑤0 < 𝜀 whenever 0 < 𝑧 − 𝑧0 < 𝛿0
And 𝑓 𝑧 − 𝑤1 < 𝜀 whenever 0 < 𝑧 − 𝑧0 < 𝛿1

4. Let 𝛿 = min (𝛿0 , 𝛿1) . Then for
0 < 𝑧 − 𝑧0 < 𝛿0
• 𝑓 𝑧 − 𝑤0 − 𝑓 𝑧 − 𝑤1 ≤ 𝑓 𝑧 − 𝑤0 + 𝑓 𝑧 − 𝑤1
< 2𝜀
ie, 𝑤1 − 𝑤0 < 2𝜀
But 𝑤1 − 𝑤0 is a constant and 𝜀 can be chosen arbitrarily small.
If we choose 𝜀 = 0 , then 𝑤1 − 𝑤0= 0
ie , 𝑤1 = 𝑤0
Hence the limit of a function is unique.

5. Continuity
• The function f ( z ) is continuous at a point 𝑧0 , if
i) lim
𝑧→𝑧0
𝑓(𝑧) exists
ii) f(𝑧0) exists
iii) lim
𝑧→𝑧0
𝑓(𝑧) = f(𝑧0)
A function of a complex variable is said to be continuous in a region R if it is
continuous at each point in R.

6. Properties
i) Sum of two continuous functions is a continuous function .
ii) Product of the continuous functions is a continuous function .
iii) Quotient of two continuous functions is continuous where the denominator
not equal to zero.
iv) Polynomial is continuous in the entire plane.

7. Derivatives
• The derivative of f at 𝑧0, denoted by 𝑓/(𝑧0) = lim
𝑧→𝑧0
𝑓 𝑧 −𝑓(𝑧0)
𝑧−𝑧0
provided this
limit exists.
• The function f is said to be differentiable at 𝑧0 when its derivative at 𝑧0
exists.

8. Analytic Functions
• A single valued function w = f ( z ) in a domain D is said to be analytic at a
point z = a in a D if there exists a neighbourhood 𝑧 − 𝑎 < 𝛿 at all points
of which the function is differentiable.
Singular Point
The point at which f ( z ) is not differentiable are called singular points of the
function.

9. Ex. Given u = 𝑦3
- 3𝑥2
y find f(z)such that f(z)
is analytic
• Solution
• dv =
𝜕𝑣
𝜕𝑥
𝑑𝑥 +
𝜕𝑣
𝜕𝑦
dy
=
−𝜕𝑢
𝜕𝑦
𝑑𝑥 +
𝜕𝑢
𝜕𝑥
dy ( C R equations)
v = (
−𝜕𝑢
𝜕𝑦
𝑑𝑥 +
𝜕𝑢
𝜕𝑥
dy )
𝜕𝑢
𝜕𝑥
= -6xy ;
𝜕𝑢
𝜕𝑦
= 3𝑦2
-3𝑥2
v = (−3𝑦2+3𝑥2)𝑑𝑥 − 6𝑥𝑦 𝑑𝑦

10. = 3𝑥2
dx – (3𝑦2
dx + 6xy dy )
= 𝑑 ( 𝑥3) - d ( 3x𝑦2 )
v = 𝑥3
- 3x𝑦2
+ c
Hence f ( z ) = u + iv = 𝑦3- 3𝑥2y + i (𝑥3 - 3x𝑦2)
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