Complex Analysis and Ordinary Differential Equations (U23MA202) - Vivekananda

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5 Subjects

Analytic Functions - VCEW

26 Exercises • 71 Learning Materials

1.1 Analytic Functions

1.1.1 Analytic function

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1.1.2 Cauchy Riemann Equation

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1.1.3 The Necessary condition for f(z) to be analytic (in Cartesian form)

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1.1.4 Sufficient condition for f(z) to be analytic

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1.1.5 Cauchy Riemann equations In polar coordinates

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1.1.6 Solved Problem 1

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5:8

1.1.6 Quiz

Exercise

1.1.7 Solved Problem 2

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6:32

1.1.7 Quiz

Exercise

1.1.8 Solved Problem 3

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5:00

1.1.8 Quiz

Exercise

1.1.9 Solved Problem 4

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1.1.10 Solved Problem 5

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1.1.11 Solved Problem 6

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1.1.12 Solved Problem 7

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1.1.13 Exercises

Exercise

1.2 Harmonic Property

1.2.1 Property of Analytic functions

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1.2.2 Harmonic Functions - Laplace Equation

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1.2.3 Harmonic Function

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1.2.4 Harmonic Conjugates

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1.2.5 Solved Problem 1

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9:54

1.2.5 Quiz

Exercise

1.2.6 Solved Problem 2

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14:58

1.2.6 Quiz

Exercise

1.2.7 Solved Problem 3

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13:19

1.2.7 Quiz

Exercise

1.2.8 Solved Problem 4

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12:36

1.2.8 Quiz

Exercise

1.2.9 Solved Problem 5

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1.2.10 Solved Problem 6

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1.2.11 Solved Problem 7

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1.2.12 Solved Problem 8

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1.2.13 Solved Problem 9

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1.2.14 Exercises

Exercise

1.3 Construction of Analytic functions

1.3.1 Construction of Analytic functions

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1.3.2 Method 1 Construction of analyti function whose real or imaginary part is known

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1.3.3 Solved Problem 1

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8:41

1.3.3 Quiz

Exercise

1.3.4 Solved Problem 2

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7:24

1.3.4 Quiz

Exercise

1.3.5 Solved Problem 3

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13:37

1.3.5 Quiz

Exercise

1.3.6 Solved problem 4

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1.3.7 Solved problem 5

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1.3.8 Exercises

Exercise

1.4 Milne Thomson Method

1.4.1 Milne Thomson Method (To Construct an analytic function)

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1.4.2 Working rule To Construct an analytic function by Milne Thomson Method

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1.4.3 Solved Problem 1

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9:51

1.4.3 Quiz

Exercise

1.4.4 Solved Problem 1

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14:43

1.4.4 Quiz

Exercise

1.4.5 Solved Problem 3

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13:23

1.4.5 Quiz

Exercise

1.4.6 Solved Problem 4

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1.4.7 Solved Problem 5

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1.4.8 Solved Problem 6

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1.4.9 Solved Problem 7

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1.4.10 Solved Problem 8

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1.4.11 Exercise

Exercise

1.5 Conformal Mapping

1.5.1 Conformal mapping

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1.5.2 Translation w=c+z

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1.5.3 Magnification w=cz

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1.5.4 Magnification and Rotation

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1.5.5 Magnification, Rotation and Translation

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1.5.6 Inversion and Reflection w=1 / z

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1.5.7 Transformation w=z^2

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1.5.8 Problem based on w=c+z _ Solved Problem 1

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4:45

1.5.8 Quiz

Exercise

1.5.9 Solved Problem 2

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1.5.10 Problem based on w=cz _ Solved Problem 3

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6:2

1.5.10 Quiz

Exercise

1.5.11 Solved Problem 4

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1.5.12 Problem based on w=1 / z _ Solved Problem 5

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6:40

1.5.12 Quiz

Exercise

1.5.13 Solved Problem 6

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1.5.14 Solved Problem 7

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1.5.15 Problem based on w=z^2 Solved Problem 8

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4:46

1.5.15 Quiz

Exercise

1.5.16 Solved Problem 9

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1.5.17 Exercises

Exercise

1.6 Bilinear Transformation (Mobius Transformation)

1.6.1 Bilinear Transformation

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1.6.2 Theorem 1

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1.6.3 Fixed points (or) Invariant points

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1.6.4 Theorem 2

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1.6.5 Cross Ratio

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1.6.6 Solved Problem 1

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7:58

1.6.6 Quiz

Exercise

1.6.7 Solved Problem 2

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6:23

1.6.7 Quiz

Exercise

1.6.8 Solved Problem 3

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5:4

1.6.8 Quiz

Exercise

1.6.9 Solved Problem 4

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1.6.10 Solved Problem 5

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1.6.11 Solved Problem 6

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1.6.12 Solved Problem 7

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1.6.13 Solved Problem 8

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1.6.14 Exercises

Exercise

Complex Integration - VCEW

22 Exercises • 79 Learning Materials

2.1 Line Integral

2.1.1 Line integral

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2.1.2 Solved Problem 1

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8:11

2.1.2 Quiz

Exercise

2.1.3 Solved Problem 2

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13:14

2.1.3 Quiz

Exercise

2.1.4 Solved Problem 3

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8:8

2.1.4 Quiz

Exercise

2.1.5 Solved Problem 4

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2.1.6 Solved Problem 5

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2.1.7 Solved Problem 6

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2.1.8 Solved Problem 7

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2.2 Cauchy's Integral Theorem and Integral Formula

2.2.1 Introduction

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2.2.2 The cauchys theorem

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2.2.3 Cauchys integral formula

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2.2.4 Generalized cauchys integral formula

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2.2.5 Some useful theorems

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2.2.6 Solved Problem 1

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7:00

2.2.6 Quiz

Exercise

2.2.7 Solved Problem 2

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2.2.8 Solved Problem 3

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9:12

2.2.8 Quiz

Exercise

2.2.9 Solved Problem 4

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11:19

2.2.9 Quiz

Exercise

2.2.10 Solved Problem 5

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2.2.11 Solved Problem 6

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2.2.12 Solved Problem 7

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2.2.13 Solved Problem 8

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2.2.14 Solved Problem 9

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2.2.15 Solved Problem 10

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2.2.16 Solved Problem 11

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2.2.17 Solved Problem 12

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2.2.18 Exercises

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2.3 Taylor’s Series

2.3.1 Introduction of complex power series

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2.3.2 Taylors series

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2.3.3 Solved Problem 1

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8:5

2.3.3 Quiz

Exercise

2.3.4 Solved Problem 2

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10:2

2.3.4 Quiz

Exercise

2.3.5 Solved Problem 3

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7:51

2.3.5 Quiz

Exercise

2.3.6 Solved Problem 4

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2.3.7 Solved Problem 5

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2.3.8 Solved Problem 6

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2.3.9 Solved Problem 7

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2.3.10 Exercises

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2.4 Laurent’s Series

2.4.1 Laurents series

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2.4.2 Solved Problem 1

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8:16

2.4.2 Quiz

Exercise

2.4.3 Solved Problem 2

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10:47

2.4.3 Quiz

Exercise

2.4.4 Solved Problem 3

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14:34

2.4.4 Quiz

Exercise

2.4.5 Solved Problem 4

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2.4.6 Solved Problem 5

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2.4.7 Solved Problem 6

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2.4.8 Exercises

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2.5 Classification of Singularities

2.5.1 Singularities and poles

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2.5.2 Solved Problem 1

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8:7

2.5.2

Exercise

2.5.3 Solved Problem 2

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8:19

2.5.3

Exercise

2.5.4 Solved Problem 3

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1:57

2.5.4

Exercise

2.5.5 Solved Problem 4

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2.5.6 Exercises

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2.6 Cauchy's Residue Theorem

2.6.1 Introduction

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2.6.2 Residues

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2.6.3 Cauchy’s residue theorem

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2.6.4 Solved Problem 1

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7:3

2.6.4

Exercise

2.6.5 Solved Problem 2

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11:6

2.6.5

Exercise

2.6.6 Solved Problem 3

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5:55

2.6.6

Exercise

2.6.7 Solved Problem 4

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9:9

2.6.7

Exercise

2.6.8 Solved Problem 5

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2.6.9 Solved Problem 6

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2.6.10 Solved Problem 7

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2.6.11 Solved Problem 8

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2.6.12 Solved Problem 9

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2.6.13 Solved Problem 10

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2.6.14 Solved Problem 11

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2.6.15 Solved Problem 12

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2.6.16 Solved Problem 13

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2.6.17 Solved Problem 14

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2.6.18 Solved Problem 15

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2.6.19 Solved Problem 16

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2.6.20 Solved Problem 17

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2.6.21 Solved Problem 18

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2.6.22 Exercises

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2.7 Integration around the Unit Circle

2.7.1 Contour integration type I

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2.7.2 Solved Problem 1

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11:59

2.7.2

Exercise

2.7.3 Solved Problem 2

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10:37

2.7.3

Exercise

2.7.4 Solved Problem 2

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10:30

2.7.4

Exercise

2.7.5 Solved Problem 4

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2.7.6 Solved Problem 5

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2.7.7 Exercises

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Vector Differentiation & Integration - VCEW

38 Exercises • 106 Learning Materials

3.1 Gradient

3.1.1 Introduction

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3.1.2 Graphical representation of vector fields

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3.1.3 Scalar and vector fields

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3.1.4 Gradient of a scalar

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3.1.5 Geometrical interpretation

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3.1.6 Solved Problem 1

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4:20

3.1.6 Quiz

Exercise

3.1.7 Solved Problem 2

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8:1

3.1.7 Quiz

Exercise

3.1.8 Solved Problem 3

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8:13

3.1.8 Quiz

Exercise

3.1.9 Solved Problem 4

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3.1.10 Solved Problem 5

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3.1.11 Solved Problem 6

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3.1.12 Solved Problem 7

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3.1.13 Solved Problem 8

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3.1.14 Exercises

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3.2 Directional Derivatives

3.2.1 Introduction

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3.2.2 Solved problem 1

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6:54

3.2.2 Quiz

Exercise

3.2.3 Solved problem 2

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3.2.4 Solved problem 3

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3.2.5 Solved problem 4

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3.2.6 Solved problem 5

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3.2.7 Solved problem 6

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3.2.8 Solved problem 7

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3.2.9 Exercises

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3.3 Curl & Divergence

3.3.1 Introduction

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3.3.2 Laplacian of a scalar field

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3.3.3 Properties

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3.3.4 Curl of a vector field

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3.3.5 Properties

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3.3.6 Vector Identities

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3.3.7 Solved Problem 1

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3:17

3.3.7 Quiz

Exercise

3.3.8 Solved Problem 2

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7:34

3.3.8 Quiz

Exercise

3.3.9 Solved Problem 3

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7:21

3.3.9 Quiz

Exercise

3.3.10 Solved Problem 4

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6:36

3.3.10 Quiz

Exercise

3.3.11 Solved Problem 5

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5:2

3.3.11 Quiz

Exercise

3.3.12 Solved Problem 6

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5:39

3.3.12 Quiz

Exercise

3.3.13 Solved Problem 7

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3.3.14 Solved Problem 8

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3.3.15 Solved Problem 9

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3.3.16 Solved Problem 10

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3.3.17 Solved Problem 11

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3.3.18 Solved Problem 12

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3.3.19 Solved Problem 13

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3.3.20 Solved Problem 14

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3.3.21 Exercises

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3.4 Solenoidal & Irrotational

3.4.1 Introduction

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3.4.2 Solved Problem 1

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3:45

3.4.2 Quiz

Exercise

3.4.3 Solved Problem 2

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2:20

3.4.3 Quiz

Exercise

3.4.4 Solved Problem 3

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3.4.5 Solved Problem 4

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3.4.6 Solved Problem 5

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3.4.7 Solved Problem 6

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3.4.8 Solved Problem 7

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3.4.9 Solved Problem 8

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3.4.10 Solved Problem 9

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3.4.11 Solved Problem 10

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3.4.12 Exercises

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3.5 Line Integral

3.5.1 Introduction

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3.5.2 Conservative vector field

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3.5.3 Solved Problem 1

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5:50

3.5.3 Quiz

Exercise

3.5.4 Solved Problem 2

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4:38

3.5.4 Quiz

Exercise

3.5.5 Solved Problem 3

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5:1

3.5.5 Quiz

Exercise

3.5.6 Solved Problem 4

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4:24

3.5.6 Quiz

Exercise

3.5.7 Solved Problem 5

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4:31

3.5.7 Quiz

Exercise

3.5.8 Solved Problem 6

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3.5.9 Solved Problem 7

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3.5.10 Solved Problem 8

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3.5.11 Solved Problem 9

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3.5.12 Exercise

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3.6 Surface Integral

3.6.1 Introduction

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3.6.2 Solved Problem 1

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8:00

3.6.2 Quiz

Exercise

3.6.3 Solved Problem 2

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3.6.4 Solved Problem 3

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3.6.5 Exercise

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3.7 Volume Integral

3.7.1 Introduction

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3.7.2 Solved Problem 1

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6:15

3.7.2 Quiz

Exercise

3.7.3 Solved Problem 2

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3.7.4 Exercise

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3.8 Greens Theorem

3.8.1 Introduction

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3.8.2 Solved Problem 1

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6:50

3.8.2 Quiz

Exercise

3.8.3 Solved Problem 2

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12:6

3.8.3 Quiz

Exercise

3.8.4 Solved Problem 3

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8:55

3.8.4 Quiz

Exercise

3.8.5 Solved Problem 4

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9:41

3.8.5 Quiz

Exercise

3.8.6 Solved Problem 5

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7:50

3.8.6 Quiz

Exercise

3.8.7 Solved Problem 6

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3.8.8 Solved Problem 7

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3.8.9 Solved Problem 8

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3.8.10 Exercise

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3.9 Gauss Divergence Theorem

3.9.1 Introduction

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3.9.2 Solved Problem 1

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15:7

3.9.2 Quiz

Exercise

3.9.3 Solved Problem 2

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5:24

3.9.3 Quiz

Exercise

3.9.4 Solved Problem 3

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4:15

3.9.4 Quiz

Exercise

3.9.5 Solved Problem 4

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6:53

3.9.5 Quiz

Exercise

3.9.6 Solved Problem 5

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13:58

3.9.6 Quiz

Exercise

3.9.7 Solved Problem 6

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8:16

3.9.7 Quiz

Exercise

3.9.8 Solved Problem 7

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10:21

3.9.8 Quiz

Exercise

3.9.9 Solved Problem 8

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3.9.10 Exercise

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3.10 Stokes Theorem

3.10.1 Introduction

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3.10.2 Solved Problem 1

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10:3

3.10.2 Quiz

Exercise

3.10.3 Solved Problem 2

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9:19

3.10.3 Quiz

Exercise

3.10.4 Solved Problem 3

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14:11

3.10.4 Quiz

Exercise

3.10.5 Solved Problem 4

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7:31

3.10.5 Quiz

Exercise

3.10.6 Solved Problem 5

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6:45

3.10.6 Quiz

Exercise

3.10.7 Solved Problem 6

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6:38

3.10.7 Quiz

Exercise

3.10.8 Solved Problem 7

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8:35

3.10.8 Quiz

Exercise

3.10.9 Exercise

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Ordinary Differential Equations - VCEW

91 Learning Materials

4.1 Linear Differential Equations of Higher Order

4.1.1 Introduction

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4.1.2 Operator D and Auxiliary Equations

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4.1.3 General solution of fDy 0

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4.1.4 Solved Problem 1

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5:39

4.1.5 Solved Problem 2

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3:32

4.1.6 Solved Problem 3

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3:53

4.1.7 Solved Problem 4

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4.1.8 Solved Problem 5

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4.1.9 Solved Problem 6

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4.1.10 Solved Problem 7

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4.1.11 Solved Problem 8

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4.1.12 Solved Problem 9

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4.1.13 Solved Problem 10

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4.1.14 Exercise

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4.2 General solution of f(D)y = e^ax

4.2.1 Introduction

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4.2.2 Particular Integral of X eax

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4.2.3 Solved Problem 1

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4:56

4.2.4 Solved Problem 2

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3:34

4.2.5 Solved Problem 3

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4:47

4.2.6 Solved Problem 4

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6:17

4.2.7 Solved Problem 5

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6:33

4.2.8 Solved Problem 6

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4.2.9 Solved Problem 7

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4.2.10 Solved Problem 8

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4.2.11 Exercise

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4.3 General solution of f(D)y = cos ax or sin ax

4.3.1 Introduction

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4.3.2 Working rule to Evaluate Particular Integral

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4.3.3 Solved Problem 1

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6:26

4.3.4 Solved Problem 2

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5:22

4.3.5 Solved Problem 3

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6:24

4.3.6 Solved Problem 4

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6:47

4.3.7 Solved Problem 5

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5:6

4.3.8 Solved Problem 6

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4.3.9 Solved Problem 7

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4.3.10 Solved Problem 8

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4.3.11 Exercise

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4.4 General solution of f(D)y = x^m

4.4.1 Introduction

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4.4.2 Solved Problem 1

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5:42

4.4.3 Solved Problem 2

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6:7

4.4.4 Solved Problem 3

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8:25

4.4.5 Solved Problem 4

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4.4.6 Solved Problem 5

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4.4.7 Solved Problem 6

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4.4.8 Solved Problem 7

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4.4.9 Solved Problem 8

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4.4.10 Exercise

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4.5 General solution of f(D)y = e^ax . f(x)

4.5.1 Introduction

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4.5.2 Solved Problem 1

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7:16

4.5.3 Solved Problem 2

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9:52

4.5.4 Solved Problem 3

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10:19

4.5.5 Solved Problem 4

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4.5.6 Solved Problem 5

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4.5.7 Exercise

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4.6 General solution of f(D)y = x^m . f(x)

4.6.1 Introduction

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4.6.2 Solved Problem 1

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8:20

4.6.3 Solved Problem 2

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8:00

4.6.4 Solved Problem 3

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7:55

4.6.5 Solved Problem 4

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4.6.6 Solved Problem 5

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4.6.7 Solved Problem 6

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4.6.8 Solved Problem 7

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4.6.9 Exercise

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4.7 Cauchy's Differential Equations

4.7.1 Introduction

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4.7.2 Solved Problem 1

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7:39

4.7.3 Solved Problem 2

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8:11

4.7.4 Solved Problem 3

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8:22

4.7.5 Solved Problem 4

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4.7.6 Solved Problem 5

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4.7.7 Solved Problem 6

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4.7.8 Solved Problem 7

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4.7.9 Solved Problem 8

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4.7.10 Solved Problem 9

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4.7.11 Exercise

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4.8 Legendre Differential Equations

4.8.1 Introduction

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4.8.2 Solved Problem 1

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7:7

4.8.3 Solved Problem 2

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6:2

4.8.4 Solved Problem 3

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7:21

4.8.5 Solved Problem 4

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4.8.6 Solved Problem 5

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4.8.7 Solved Problem 6

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4.8.8 Solved Problem 7

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4.8.9 Solved Problem 8

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4.8.10 Exercise

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4.9 Method of variation of parameters

4.9.1 Introduction

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4.9.2 Solved Problem 1

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6:48

4.9.3 Solved Problem 2

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7:42

4.9.4 Solved Problem 3

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10:13

4.9.5 Solved Problem 4

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4.9.6 Solved Problem 5

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4.9.7 Solved Problem 6

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4.9.8 Exercise

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Laplace Transforms - VCEW

136 Learning Materials

5.1 Laplace Transform

5.1.1 Introduction

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5.1.2 Sufficient Conditions for Existence

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5.1.3 Solved Problem 1

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5.1.4 Exponential Order

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5.2 Laplace Transform of Some Standard Functions

5.2.1 Introduction

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5.2.2 Laplace Transform of e^at

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5:12

5.2.3 Laplace Transform of 1

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3:54

5.2.4 Laplace Transform of sinat and cosat

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5:28

5.2.5 Laplace Transform of sinhat and coshat

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7:20

5.2.6 Laplace Transform of t^n

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4:52

5.2.7 Solved Problem 1

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6:15

5.2.8 Solved Problem 2

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8:2

5.2.9 Solved Problem 3

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8:36

5.2.10 Solved Problem 4

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5.2.11 Solved Problem 5

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5.2.12 Solved Problem 6

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5.2.13 Solved Problem 7

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5.2.14 Exercises

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5.3 First, Second Shifting Theorem and Change of Scale Property

5.3.1 First Shifting Theorem

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5.3.2 Solved Problem 1

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5:39

5.3.3 Solved Problem 2

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5:16

5.3.4 Solved Problem 3

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3:49

5.3.5 Solved Problem 4

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5.3.6 Solved Problem 5

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5.3.7 Second Shifting Theorem

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5.3.8 Solved Problem 1

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4:8

5.3.9 Solved Problem 2

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5.3.10 Solved Problem 3

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5.3.11 Change of Scale Property

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5.3.12 Solved Problem 1

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5:37

5.3.13 Solved Problem 2

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5.3.14 Solved Problem 3

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5.3.15 Exercises

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5.4 Transforms of Derivatives

5.4.1 Transforms of Derivatives

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5.4.2 Derivatives of Transforms

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5.4.3 Solved Problem 1

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10:2

5.4.4 Solved Problem 2

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7:43

5.4.5 Solved Problem 3

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5.4.6 Solved Problem 4

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5.4.7 Solved Problem 5

Video

7:34

5.4.8 Exercises

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5.5 Transforms of Integrals

5.5.1 Transforms of Integrals

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5.5.2 Integrals of Transforms

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5.5.3 Solved Problem 1

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10:00

5.5.4 Solved Problem 2

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10:53

5.5.5 Solved Problem 3

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10:52

5.5.6 Solved Problem 4

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5.5.7 Solved Problem 5

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5.5.8 Solved Problem 6

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5.5.9 Solved Problem 7

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5.5.10 Solved Problem 8

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5.5.11 Solved Problem 9

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5.5.12 Solved Problem 10

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5.5.13 Exercises

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5.6 Laplace Transform of Periodic Functions

5.6.1 Laplace Transform of Periodic Functions

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5.6.2 Solved Problem 1

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6:56

5.6.3 Solved Problem 2

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12:12

5.6.4 Solved Problem 3

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5.6.5 Solved Problem 4

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5.6.6 Solved Problem 5

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5.6.7 Exercises

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5.7 Laplace Transform of Unit Step and Unit Impulse Functions

5.7.1 Laplace Transform of Unit Step Functions

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5.7.2 Laplace Transform of Unit Impulse Functions

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5.7.3 Solved Problem 1

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4:28

5.7.4 Solved Problem 2

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12:39

5.7.5 Solved Problem 3

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11:17

5.7.6 Solved Problem 4

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5.7.7 Solved Problem 5

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5.7.8 Exercises

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5.8 Inverse Laplace Transform

5.8.1 Inverse Laplace Transform

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5.8.2 Solved Problem 1

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3:40

5.8.3 Solved Problem 2

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6:2

5.8.4 Solved Problem 3

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5.8.5 Solved Problem 4

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5.8.6 Solved Problem 5

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5.8.7 Method of Partial Fraction

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5.8.8 Solved Problem 3

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6:4

5.8.9 Solved Problem 3

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5.8.10 Solved Problem 4

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5.8.11 Solved Problem 5

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5.8.12 Solved Problem 6

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5.8.13 Real time problem

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5.8.14 Real time problem

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5.8.15 Exercises

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5.9 First and Second Shifting Theorem, Change of Scale of Property

5.9.1 First Shifting Theorem

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5.9.2 Solved Problem 2

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7:9

5.9.3 Solved Problem 3

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5.9.4 Second Shifting Theorem

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5.9.5 Solved Problem 1

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4:47

5.9.6 Solved Problem 2

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5.9.7 Solved Problem 3

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5.9.8 Change of Scale of Property

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5.9.9 Solved Problem 1

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4:49

5.9.10 Solved Problem 2

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5.9.11 Solved Problem 3

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5.9.12 Exercises

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5.10 Inverse Laplace Transform of Derivatives and Integrals

5.10.1 Inverse Laplace Transform of Derivatives

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5.10.2 Solved Problem 1

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6:14

5.10.3 Solved Problem 2

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5.10.4 Solved Problem 3

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5.10.5 Solved Problem 4

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5.10.6 Solved Problem 5

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5.10.7 Inverse Laplace Transform of Integrals

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5.10.8 Solved Problem 1

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4:25

5.10.9 Solved Problem 2

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6:9

5.10.10 Solved Problem 3

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5.10.11 Solved Problem 4

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5.10.12 Exercises

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5.11 Multiplication by Power of s and Division by s

5.11.1 Multiplication by Power of ‘s’

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5.11.2 Solved Problem 1

Video

4:2

5.11.3 Solved Problem 2

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5.11.4 Solved Problem 3

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5.11.5 Division by ‘s’

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5.11.6 Solved Problem 2

Video

2:1

5.11.7 Solved Problem 3

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6:30

5.11.8 Exercises

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5.12 Convolution theorem

5.12.1 Convolution theorem

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5.12.2 Solved Problem 1

Video

7:12

5.12.3 Solved Problem 2

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8:29

5.12.4 Solved Problem 3

Video

8:28

5.12.5 Solved Problem 4

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5.12.6 Solved Problem 5

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5.12.7 Solved Problem 6

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5.12.8 Solved Problem 7

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5.12.9 Solved Problem 8

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5.12.10 Exercises

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5.13 Solving ordinary equations by Laplace Transforms

5.13.1 Introduction

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5.13.2 Solved Problem 1

Video

10:22

5.13.3 Solved Problem 2

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10:2

5.13.4 Solved Problem 3

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10:48

5.13.5 Solved Problem 5

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5.13.6 Solved Problem 6

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5.13.7 Solved Problem 7

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5.13.8 Solved Problem 8

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5.13.9 Solved Problem 9

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5.13.10 Exercises

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Course Instructor

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