This document is a report on Complex Variable & Numerical Method prepared by 5 students and guided by Miss. Chaitali Shah. It contains the following topics: complex numbers, complex variable, basic definitions, limits, continuity, differentiability, analytic functions, Cauchy-Riemann equations, harmonic functions, Milne-Thomson method, and applications of complex functions/variables in engineering. Examples are provided to illustrate several concepts.

1. Complex Variable & Numerical Method (2141905)
B.E. MECH – Sem IV th
Prepared by,
Patel Jeet R (170953119029)
Patel Meet (170953119030)
Patel Shrey B (170953119031)
Pathak Jeet (170953119032)
Patil Gaurav (170953119033)
Guided by,
Miss. Chaitali Shah
(Applied Mathematics Deptt.)

2. content
 Complex Number
 Complex Variable
 Basic Defination
 Limits
 Continuity
 Differentiability
 Analytic Function
 Necessary condition for f(z) CR Equation
 Sufficient Condition for f(z) to be analytic
 Polar form of CR Equation
 Harmonic Function
 Propertied of Analytic Function
 Milne-Thomson Method
 Application of complex
Function/Variable in
Engineering
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3. COMPLEX NUMBER
WHY WE STUDY ?
For most human tasks, real numbers (or even rational numbers) offer an adequate
description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones,
but essential to a person comparing the sizes of different collections of stones. Negative numbers
such as -3 and -5 are meaningless when measuring the mass of an object, but essential when
keeping track of monetary debits and credits.
All numbers are imaginary (even "zero" was contentious once). Introducing the square
root(s) of minus one is convenient because (i) all n-degree polynomials with real coefficients
then have n roots, making algebra "complete"; (ii) it saves using matrix representations for
objects that square to -1 (such objects representing an important part of the structure of linear
equations which appear in quantum mechanics, heat diffusion, optics, etc). The hottest
contenders for numbers without purpose are probably the p-adic numbers (an extension of the
rationals), and perhaps the expiry dates on army ration packs
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4. COMPLEX NUMBER
INTRODUCTION
- Complex Number is defined as an ordered pair of real number 𝑥 & 𝑦 and is denoted by (𝑥, 𝑦)
- It is also written as 𝒛 = 𝒙, 𝒚 = 𝒙 + 𝒊𝒚, where 𝑖2 = −1
𝑥 is called Real Part of z and written as Re(z)
y is called imaginary part of z and written as Im(z).
- If R(z) = 0 then 𝑧 = 𝑖𝑦 , is called Purely Imaginary Number.
- If I(z) = 0 then 𝑧 = 𝑥 , is called Purely Real Number.
- Here 𝑖 can be written as (0, 1) = 0 ± 1𝑖
Note :- −𝒂 = 𝑎 −1 = 𝑖 𝑎
- If 𝑧 = 𝑥 + 𝑖𝑦 is complex number then its conjugate or complex conjugate is defined as
𝒛 = 𝒙 − 𝒊𝒚.
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5. COMPLEX NUMBER
DIFFERENT FORMS OF COMPLEX NUMBER
 Cartesian or Rectangular Form :- 𝑧 = 𝑥 + 𝑖𝑦
 Polar Form :- 𝑧 = 𝑟(cos 𝜃 + 𝑖 sin 𝜃) 𝑜𝑟 𝑧 = 𝑟 ∠𝜃
 Exponential Form :- 𝑧 = 𝑟 𝑒 𝑖𝜃
MODULUS & ARGUMENT OF COMPLEX NUMBER
 Modulus of complex number (|z|) OR mod(z) OR 𝑟 = 𝑥2 + 𝑦2
 Argument OR Amplitude of complex number (𝜃) OR arg 𝑧 OR amp z = tan−1 𝑦
𝑥
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6. COMPLEX FUNCTION
A function whose range is in complex number, is said to be complex function OR
complex Valued Function.
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COMPLEX VARIABLE
A symbol such as 𝑧, which can stand for any one of set ogf complex number is
called as Complex Variable.

7. BASIC DEFINATION
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8. BASIC DEFINATION
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9. LIMIT OF FUNCTION OF COMPLEX VARIABLE
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10. LIMIT OF FUNCTION OF COMPLEX VARIABLE
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EXAMPLE
(1) Find Following Limit,
𝒍𝒊𝒎
𝒛 → 𝒊
𝒛 𝟐+𝟏
𝒛 𝟔+𝟏
- As 𝑧 → 𝑖, 𝑧2 → −1
∴
𝑙𝑖𝑚
𝑧 → 𝑖
𝑧2+1
𝑧2+1 (𝑧4−𝑧2+1)
=
𝑙𝑖𝑚
𝑧2 → −1
1
𝑧4−𝑧2+1
=
1
3

11. LIMIT OF FUNCTION OF COMPLEX VARIABLE
(2) Show that
𝒍𝒊𝒎
𝒛 → 𝟎
𝒛
|𝒛|
does not exist.
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12. CONTINUITY
CONTINUITY IN TERMS OF REAL & IMAGINARY NUMBER :-
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13. CONTINUITY
EXAMPLE
(1)
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14. CONTINUITY
(2)
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15. $ɧƦɛƴ ´ƶ 15
CONTINUITY

16. DIFFERENTIABILITY
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17. (1) Consider function 𝒇 𝒛 = 𝟒𝒙 + 𝒚 + 𝒊(𝟒𝒚 − 𝒙) and discuss
𝒅𝒇
𝒅𝒛
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DIFFERENTIABILITY

18. $ɧƦɛƴ ´ƶ 18
DIFFERENTIABILITY

19. DIFFERENTIABILITY
(2)
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20. ANALYTIC FUNCTION
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21. ANALYTIC FUNCTION
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THE NECESSARY CONDITION FOR 𝒇(𝒛) TO BE ANALYTIC

22. ANALYTIC FUNCTION
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23. $ɧƦɛƴ ´ƶ 23
ANALYTIC FUNCTION

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THE SUFFICIENT CONDITION FOR 𝒇(𝒛) TO BE ANALYTIC
ANALYTIC FUNCTION

25. EXAMPLE
(1)
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ANALYTIC FUNCTION

26. ANALYTIC FUNCTION
(2)
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27. C-R EQUATION IN POLAR FORM
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28. $ɧƦɛƴ ´ƶ 28
C-R EQUATION IN POLAR FORM

29. HARMONIC FUNCTION
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A real function of 𝜙 of two variables 𝑥 & 𝑦 is said to be harmonic function in region R , if it
has continuous second order partial derivatives and satisfies Lapalce Equation,
𝝏 𝟐 𝝓
𝝏𝒙 𝟐
+
𝝏 𝟐 𝝓
𝝏𝒚 𝟐
= 𝟎

30. HARMONIC FUNCTION
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31. HARMONIC FUNCTION
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32. MLINE THMSON METHOD
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33. MLINE THMSON METHOD
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34. MLINE THMSON METHOD
EXAMPLE
(1)
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35. MLINE THMSON METHOD
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36. APPLICATION OF COMPLEX FUNCTION
1. Solutions to 2-D Laplace equation by means of
conformal mapping.
2. Quantum mechanics.
3. Series expansions with analytic continuation.
4. Transformation between special functions,
5. Contour integrals :
a) Evaluate definite integrals & series.
b)Invert power series.
c) Form infinite products.
d)Asymptotic solutions.
e) Stability of oscillations.
f) Invert integral transforms.
6. Control Theory
7. Signal analysis
8. Fluid Dynamics
9. Electromagnetism:
10. 2-D designing of buildings and cars
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37. APPLICATION OF COMPLEX FUNCTION
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