Lovely Professional University, Punjab
Course Code MTH251 Course Category Course Title FUNCTION OF COMPLEX VARIABLE AND TRANSFORM Courses with Numerical focus Course Planner 16423::Harsimran Kaur Lectures 3.0 Tutorials Practicals Credits 2.0 0.0 4.0

TextBooks Sr No T-1 Title Advanced Engineering Mathematics Reference Books Sr No R-1 R-2 Other Reading Sr No OR-1 Journals articles as Compulsary reading (specific articles, complete reference) Journals atricles as compulsory readings (specific articles, Complete reference) , Title Higher Engineering Mathematics Advanced Modern Engineering Mathematics Author Grewal, B. S. Glyn James Edition 40th 3rd Year 2007 2011 Publisher Name Khanna Publishers Pearson Author Jain R. K. and Iyenger S. R. K. Edition 3rd Year 2007 Publisher Name Narosa

Relevant Websites Sr No RW-1 RW-2 (Web address) (only if relevant to the course) www2.latech.edu/~schroder/comp_var_videos.htm freescienceonline.blogspot.com/2010_04_01_archive.html Salient Features Topic videos available Complex Analysis Reference Material Available

LTP week distribution: (LTP Weeks) Weeks before MTE Weeks After MTE Spill Over 7 6 2

Detailed Plan For Lectures

Week Number

Lecture Number

Broad Topic(Sub Topic)

Chapters/Sections of Text/reference books

Other Readings, Lecture Description Relevant Websites, Audio Visual Aids, software and Virtual Labs Introduction Functions of a Complex Variable

Learning Outcomes

Pedagogical Tool Demonstration/ Case Study / Images / animation / ppt etc. Planned Lecturing

Week 1

Lecture 1

Functions of a Complex Variable and Analyticity(Introduction)

T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.4

Knowledge of Basics

Functions of a Complex Variable T-1:Ch 10 Functions and Analyticity(Domain & Range) of a Complex Variable Analytic Functions Topic 10.4 Functions of a Complex Variable and Analyticity(Single and Multi Valued Functions) T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.4 T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.7 R-1:Ch 20 Calculus of Complex Functions Topic 20.4 T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.7.1

Introduction Functions of a Complex Variable

Knowledge of Basics

Lecturing

Introduction Functions of a Complex Variable

Knowledge of Basics

Lecturing

Lecture 2

Functions of a Complex Variable and Analyticity(Analytic Functions)

Analytic Functions and CR Equations

Development of Concept

White Board Teaching

Functions of a Complex Variable and Analyticity(CR Equations)

Analytic Functions and CR Equations Necessary and sufficient conditions for function to be analytic Analytic Functions and CR Equations Necessary and sufficient conditions for function to be analytic

Development of Concept

White Board Teaching

Lecture 3

Functions of a Complex Variable and Analyticity(CR Equations)

T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.7.1

Development of Concept

White Board Teaching

Functions of a Complex Variable T-1:Ch 10 Functions and Analyticity(Necessary and of a Complex Sufficient conditions for f(z) to be Variable Analytic analytic) Functions Topic 10.7.1

Necessary and sufficient Development of conditions for function to Concept be analytic

White Board Teaching

Week 2

Lecture 4

Analyticity(Contd.), Harmonic Functions And Complex Integration(Polar form of CR equations)

T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.7.1 T-1:Ch 10 Functions of a Complex Variable Analytic Functions Topic 10.8 R-1:Ch 20 Calculus of Complex Functions Topic 20.5(1) T-1:Ch 11 Integration of Complex Functions Topics11.1 11.2 R-1:Ch 20 Calculus of Complex Functions Topic 20.12 T-1:Ch 11 Integration of Complex Functions Topics11.1 11.2 R-1:Ch 20 Calculus of Complex Functions Topic 20.12 T-1:Ch 11 Integration of Complex Functions Topic 11.2.2 T-1:Ch 11 Integration of Complex Functions Topic 11.3 T-1:Ch 11 Integration of Complex Functions Topic 11.3

Polar form of CR Equations

Numerical Applicability

White Board Teaching

Lecture 5

Analyticity(Contd.), Harmonic Functions And Complex Integration(Harmonic functions)

Harmonic functions

Development of Concept

White Board Teaching

Lecture 6

Analyticity(Contd.), Harmonic Functions And Complex Integration(Definite Integrals)

Introduction Integration Knowledge of Basics of Complex Functions Definite Integrals

Lecturing

Week 3

Lecture 7

Analyticity(Contd.), Harmonic Functions And Complex Integration(Definite Integrals)

Introduction Integration Knowledge of Basics of Complex Functions Definite Integrals

Lecturing

Lecture 8

Analyticity(Contd.), Harmonic Functions And Complex Integration(Contour Integrals)

Contour Integrals

Development of Concept

White Board Teaching

Lecture 9

Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Simply Connected and Multiply Connected Domains) Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Cauchys Integral Theorem)

Simply Connected and Multiply Connected Domains

Knowledge of Basics

White Board Teaching

Week 4

Lecture 10

Cauchys Integral Theorem

Development of Concept

White Board Teaching

Week 4

Lecture 11

Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Cauchys Integral Formula) Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Taylor Series)

R-1:Ch 20 Calculus of Complex Functions Topic 20.14 T-1:Ch 12 Power Series Taylor and Laurent Series Topic 12.5 R-1:Ch 20 Calculus of Complex Functions Topic 20.16(1) T-1:Ch 12 Power Series Taylor and Laurent Series Topic 12.6 R-1:Ch 20 Calculus of Complex Functions Topic 20.16(1) T-1:Ch 13 Zeroes Singularities and Residues Topics 13.1 13.2 R-1:Ch 20 Calculus of Complex Functions Topic 20.17(1) T-1:Ch 13 Zeroes Singularities and Residues Topic 13.3 T-1:Ch 13 Zeroes Singularities and Residues Topic 13.4 R-1:Ch 20 Calculus of Complex Functions Topics 20.18(2) 20.19

Cauchys Integral Formula

Numerical Applicability

White Board Teaching

Lecture 12

Taylor Series

Numerical Applicability

White Board Teaching

Week 5

Lecture 13

Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Laurent Series)

Laurent Series

Numerical Applicability

White Board Teaching

Lecture 14

Cauchys Theorem, Cauchys Integral Formula, Power Series, Zeroes and Singularities(Zeroes and Singularities of a Complex Function)

Introduction Zeroes and Knowledge of Basics Singularities of a Complex Function

Lecturing

Lecture 15

Residue Theorem, Applications and Transformations(Residues)

Residues

Development of Concept

White Board Teaching

Week 6

Lecture 16

Residue Theorem, Applications and Transformations(Residue Theorem)

Residue Theorem

Development of Concept

White Board Teaching

Week 6

Lecture 17

Residue Theorem, Applications T-1:Ch 13 Zeroes and Transformations(Evaluation of Singularities and Real Integrals using Residues) Residues Topics 13.5 R-1:Ch 20 Calculus of Complex Functions Topic 20.20 Residue Theorem, Applications T-1:Ch 13 Zeroes and Transformations(Evaluation of Singularities and Real Integrals using Residues) Residues Topics 13.5 R-1:Ch 20 Calculus of Complex Functions Topic 20.20 Residue Theorem, Applications and Transformations(Introduction to Transformations) T-1:Ch 14 Bilinear Transformations and Conformal Mapping Topics 14.1 14.2.1 14.2.2 R-1:Ch 20 Calculus of Complex Functions Topics 20.8(1)(2)(3) T-1:Ch 14 Bilinear Transformations and Conformal Mapping Topics 14.1 14.2.1 14.2.2 R-1:Ch 20 Calculus of Complex Functions Topics 20.8(1)(2)(3)

Evaluation of Real Numerical Integrals using Residues Applicability

White Board Teaching

Lecture 18

Evaluation of Real Numerical Integrals using Residues Applicability

White Board Teaching

Week 7

Lecture 19

Introduction to Transformations

Knowledge of Basics

Lecturing

Residue Theorem, Applications and Transformations(Linear and Inverse Transformations)

Introduction to Transformations

Knowledge of Basics

Lecturing

Bilinear Transformations(Standard T-1:Ch 14 Bilinear Transformations) Transformations and Conformal Mapping Topics 14.1 14.2.1 14.2.2 R-1:Ch 20 Calculus of Complex Functions Topics 20.8(1)(2)(3)

Introduction to Transformations

Knowledge of Basics

Lecturing

Week 7

Lecture 20

Bilinear Transformations(Invariant T-1:Ch 14 Bilinear Points) Transformations and Conformal Mapping Topics 14.3 R-1:Ch 20 Calculus of Complex Functions Topics 20.8(4) Bilinear Transformations (Introduction) T-1:Ch 14 Zeroes Singularities and Residues Topics 14.3 R-1:Ch 20 Calculus of Comlex Functions Topics 20.8(4) T-1:Ch 14 Zeroes Singularities and Residues Topics 14.3 R-1:Ch 20 Calculus of Complex Functions Topics 20.8(4) T-1:Ch 14 Bilinear Transformations and Conformal Mapping Topics 14.3

Bilinear Transformations Development of Concept

White Board Teaching

Bilinear Transformations Knowledge of Basics

Lecturing

Bilinear Transformations(Bilinear Transformations)

Bilinear Transformations Knowledge of Basics

Lecturing

Lecture 21

Bilinear Transformations(w = f(z))

Transformation w fz

Development of Concept

White Board Teaching

MID-TERM
Week 8 Lecture 22 Partial Differential Equations (Introduction) R-1:Ch 17 Partial Differential Equations Topics 17.1 17.2 Partial Differential Equations Introduction,Formation of Partial Differential Equations Partial Differential Equations Introduction,Formation of Partial Differential Equations Knowledge of Basics Lecturing

Partial Differential Equations (Formation of Partial Differential Equations)

R-1:Ch 17 Partial Differential Equations Topics 17.1 17.2

Development of Concept

White Board Teaching

Lecture 23

Partial Differential Equations (Solutions of Partial Differential Equations)

R-1:Ch 17 Partial Differential Equations Topics 17.3 17.4 17.5

Solutions of Partial Numerical Differential Equations, Applicability Linear Equations of First Order

White Board Teaching

Week 8

Lecture 23

Partial Differential Equations (Equations solvable by Direct Integration)

R-1:Ch 17 Partial Differential Equations Topics 17.3 17.4 17.5 R-1:Ch 17 Partial Differential Equations Topics 17.3 17.4 17.5 R-1:Ch 17 Partial Differential Equations Topics 17.8 to 17.11 R-1:Ch 18 Applications of Partial Differential Equations Topics 18.1 18.2

Solutions of Partial Numerical Differential Equations, Applicability Linear Equations of First Order Solutions of Partial Numerical Differential Equations, Applicability Linear Equations of First Order Homogeneous Linear Numerical Equations with Constant Applicability Coefficients

White Board Teaching

Partial Differential Equations (Linear Equations of First Order)

White Board Teaching

Lecture 24

Partial Differential Equations (Homogeneous Linear Equations with Constant Coefficients)

White Board Teaching

Week 9

Lecture 25

Applications of Partial Differential Equations(Introduction)

Applications of Partial Differential Equations Introduction,Method of Separation of Variables

Knowledge of Basics

Lecturing

Applications of Partial Differential R-1:Ch 18 Equations(Method of Separation of Applications of Variables) Partial Differential Equations Topics 18.1 18.2 Lecture 26 Applications of Partial Differential Equations(Wave Equation) R-1:Ch 18 Applications of Partial Differential Equations Topics 18.3 18.4 18.4(2) R-1:Ch 18 Applications of Partial Differential Equations Topics 18.5(1)(2) 18.6 R-1:Ch 18 Applications of Partial Differential Equations Topics 18.5(1)(2) 18.6

Applications of Partial Differential Equations Introduction,Method of Separation of Variables

Numerical White Board Teaching ApplicabilityNumerical Applicability

Wave Equation

Development of Concept

White Board Teaching

Lecture 27

Applications of Partial Differential Equations(Heat Equation)

One Dimensional Heat Flow Two Dimensional Heat Flow One Dimensional Heat Flow Two Dimensional Heat Flow

Development of Concept

White Board Teaching

Week 10

Lecture 28

Applications of Partial Differential Equations(Heat Equation)

Development of Concept

White Board Teaching

Week 10

Lecture 29

Applications of Partial Differential Equations(Laplace Equation)

R-1:Ch 18 Applications of Partial Differential Equations Topics 18.7 T-1:Ch 8 Laplace Transformation Topics 8.1 R-1:Ch 21 Laplace Transforms Topics 21.1 21.2 T-1:Ch 8 Laplace Transformation Topics 8.1 R-1:Ch 21 Laplace Transforms Topics 21.1 21.2 T-1:Ch 8 Laplace Transformations Topic 8.2 R-1:Ch 21 Laplace Transforms Topic 21.3 R-1:Ch 21 Laplace Transforms Topics 21.12 T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Transforms 9.1 9.6 R-1:Ch 22 Fourier Transforms Topics 22.1 22.2 22.4 T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Transforms Topics 9.1 9.6 R-1:Ch 22 Fourier Transforms Topics 22.1 22.2 22.4

Laplace Equation

Development of Concept

White Board Teaching

Lecture 30

Laplace Transformations (Introduction)

Laplace Transformations Knowledge of Basics Introduction

Lecturing

Laplace Transformations (Definition)

Laplace Transformations Knowledge of Basics Introduction

Lecturing

Week 11

Lecture 31

Laplace Transformations (Transforms of Elementary Functions)

Transforms of Elementary Functions

Knowledge of Basics

Lecturing

Lecture 32

Laplace Transformations(Inverse Transforms by method of Partial Fractions) Fourier Transforms(Introduction)

Inverse Transforms by method of Partial Fractions Fourier Transforms Introduction, Definition

Numerical Applicability

White Board Teaching

Lecture 33

Knowledge of Basics

Lecturing

Fourier Transforms(Definition)

Fourier Transforms Introduction, Definition

Knowledge of Basics

Lecturing

Week 12

Lecture 34

Fourier Transforms(Fourier Sine and Cosine Transforms)

T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Transforms Topics 9.6 R-1:Ch 22 Fourier Transforms Topics 22.4 T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Transforms Topics 9.6 R-1:Ch 22 Fourier Transforms Topics 22.6 R-1:Ch 22 Fourier Transforms Topics 22.8 R-2:Ch 3 The Z Transform Topic 3.7 T-1:Ch 17 Z Transformation Topics 17.1 17.2 R-1:Ch 23 Z Transforms Topics 23.1 23.2 23.3 R-2:Ch 3 The Z Transform Topics 3.1 3.2.1 T-1:Ch 17 Z Transformation Topics 17.1 17.2 R-1:Ch 23 Z Transforms Topics 23.1 23.2 23.3 R-2:Ch 3 The Z Transform Topics 3.1 3.2.1

Fourier Sine and Cosine Development of Transforms Concept

White Board Teaching

Lecture 35

Fourier Transforms(Convolution Theorem)

Convolution Theorem

Development of Concept

White Board Teaching

Lecture 36

Fourier Transforms(Relation between Fourier and Laplace Transforms)

Relation between Fourier Development of and Laplace Transforms Concept

White Board Teaching

Week 13

Lecture 37

Z-Transforms(Introduction)

ZTransforms Knowledge of Basics Introduction,Definition,Z Transforms of Standard functions

Lecturing

Z-Transforms(Definition)

ZTransforms Knowledge of Basics Introduction,Definition,Z Transforms of Standard functions

Lecturing

Week 13

Lecture 37

Z-Transforms(Z-Transforms of Standard functions)

T-1:Ch 17 Z Transformation Topics 17.1 17.2 R-1:Ch 23 Z Transforms Topics 23.1 23.2 23.3 R-2:Ch 3 The Z Transform Topics 3.1 3.2.1 T-1:Ch 17 Z Transformation Topics 17.2 R-1:Ch 23 Z Transforms Topics 23.15 R-2:Ch 3 The Z Transform Topics 3.4 T-1:Ch 17 Z Transformation Topics 17.3 R-1:Ch 23 Z Transforms Topics 23.16

ZTransforms Development of Introduction,Definition,Z Concept Transforms of Standard functions

White Board Teaching

Lecture 38

Z-Transforms(Inverse ZTransforms)

Inverse ZTransforms

Development of Concept

White Board Teaching

Lecture 39

Z-Transforms(Applications to Differential Equations)

Applications to Differential Equations

Numerical Applicability

White Board Teaching

SPILL OVER
Week 14 Lecture 40 T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Tansforms Topic 9.5.3 T-1:Ch 9 Fourier Series Fourier Integrals and Fourier Tansforms Topic 9.5.4 Fourier Series Solution of Heat Equation Numerical Applicability Lecturing

Lecture 41

Fourier Series Solution of Wave Equation

Numerical Applicability

Lecturing

Scheme for CA:
Component Test Frequency 2 Total :Out Of 3 Each Marks Total Marks 10 10 20 20

Details of Academic Task(s)
AT No. Objective Topic of the Academic Task Nature of Academic Task (group/individuals/field work Evaluation Mode Allottment / submission Week 3/4

Test 1

To test analytical Introduction - Functions of a Complex Variable, Analytic Individual skills of the students Functions and CR Equations, Necessary and sufficient conditions about the topic. for function to be analytic, Polar form of CR Equations, Harmonic functions, Introduction - Integration of Complex Functions, Definite Integrals, Contour Integrals, Simply Connected and Multiply Connected Domains, Cauchys Integral Theorem To test analytical Cauchys Integral Formula, Taylor Series, Laurent Series, skills of the students Introduction - Zeroes and Singularities of a Complex Function, about the topic. Residues, Residue Theorem Individual

As per guidelines in Teachers Manual

Test 2

As per guidelines in Teachers Manual

5/6

Test 3

To test analytical Partial Differential Equations - Introduction,Formation of Partial Individual skills of the students Differential Equations, Solutions of Partial Differential Equations, about the topic. Linear Equations of First Order, Homogeneous Linear Equations with Constant Coefficients, Applications of Partial Differential Equations : Introduction,Method of Separation of Variables, Wave Equation, One Dimensional Heat Flow, Two Dimensional Heat Flow, Laplace Equation, Laplace Transformations Introduction, Transforms of Elementary Functions

As per guidelines in Teachers Manual

10 / 11

Plan for Tutorial: (Please do not use these time slots for syllabus coverage)
Tutorial No. Lecture Topic Type of pedagogical tool(s) planned (case analysis,problem solving test,role play,business game etc)

Tutorial 1

Problems on domain and range of functions of a complex variable Problems on Analyticity and CR Equations Problems on CR Equations and polar form of CR equations Problems on Harmonic Functions Problems on Definite Integrals Problems on Contour Integrals

Problem solving

Tutorial 2 Tutorial 3

Problem solving Problem solving

Tutorial 4 Tutorial 5 Tutorial 6

Problem solving Problem solving Problem solving

Tutorial 7

Problems on Cauchy's Integral Theorem and Cauchy's Integral Formula Test 1 Different Tests for Separate Groups Problems on Taylor's Series Problems on Laurent's Series and on singularities and poles Problems on Residues and on Evaluation of Real Integrals Test 2 Different Tests for Separate Groups Problems on Inverse Transformations, Fixed points Problems on Bilinear Transformations and w = f(z)

Problem solving

Tutorial 8 Tutorial 9 Tutorial 10

Test Problem solving Problem solving

Tutorial 11

Problem solving

Tutorial 12 Tutorial 13 Tutorial 14

Test Problem solving Problem solving

After Mid-Term
Tutorial 15 Problems on Solutions of PDE and Linear Equations of Problem solving First order Problems on Homogeneous Linear Equations with Constant Coefficients Problems on Separation of Variables Problems on Wave Equation Problems on Heat Equation Problems on Laplace Equation Problems on Laplace Transforms of Elementary Functions, on Properties and on Inverse Laplace Transforms Test 3 Different Tests for Separate Groups Problems on Fourier Sine and Cosine Transforms Problems on Convolution Theorem and Relation between Fourier and Laplace Transforms Problem solving

Tutorial 16

Tutorial 17 Tutorial 18 Tutorial 19 Tutorial 20 Tutorial 21

Problem solving Problem solving Problem solving Problem solving Problem solving

Tutorial 22

Test

Tutorial 23 Tutorial 24

Problem solving Problem solving

Tutorial 25

Problems on Z Transforms of Standard Functions and Inverse Z Transforms Problems on Applications to Differential Equations

Problem solving

Tutorial 26

Problem solving