23MAT102

COMPLEX ANALYSIS AND LAPLACE TRANSFORMS

L

T

P

J

C

(Common to AEROSPACE,AGRI,AUTO,BME,CIVIL,ECE,EEE,FT,MECH,MCT)

3

1

0

0

4

UNIT I

VECTOR CALCULUS                                                                                        

9+3

Derivatives: Gradient and Directional derivatives – Divergence and Curl of a vector field – Solenoidal and Irrotational of a vector – Green’s, Gauss divergence and Stoke’s theorems (statements only) – Verification of theorems and application in evaluating line, surface and volume integrals.

UNIT II

ORDINARY DIFFERENTIAL EQUATIONS

9+3

Higher order linear differential equations with constant coefficients – Method of variation of parameters – Homogenous equation of Euler’s and Legendre’s type – Solution of system of simultaneous linear first order differential equations with constant coefficients.

UNIT III

COMPLEX DIFFERENTIATION

9+3

Derivatives of f(z) – Analytic function – Cauchy-Riemann Equations – Harmonic function – Harmonic conjugate – Construction of Analytic function  – Conformal Mapping – Mapping by functions: w = z + c, cz, 1/z  – Bilinear transformations  –  Application to flow problems.

UNIT IV

COMPLEX INTEGRATION

9+3

Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s series – Zeros of an analytic function – singularities – Laurent’s series – Residues – Cauchy Residue theorem .

UNIT V

LAPLACE TRANSFORMS

9+3

Definition, properties, existence conditions – Transforms of elementary functions – Shifting theorem – Transforms of derivatives and integrals – Periodic functions – Initial and final value theorem – Inverse  transforms – Application to solution of  linear second order ordinary differential equations with constant coefficients.

L : 45

T:15

P: 0

J: 0

Total: 60 PERIODS

TEXT BOOKS

1.

Kreyszig.E, Advanced Engineering Mathematics, John Wiley and Sons, 10^th Edition, New Delhi 2016.

2.

Grewal.B.S., Higher Engineering Mathematics, Khanna Publishers, New Delhi, 44^th Edition, 2018.

REFERENCES

1.

Bali. N.P, Goyal. M. and Watkins. C., Advanced Engineering Mathematics, Firewall Media (An imprint of Lakshmi Publications Pvt., Ltd.,), New Delhi, 7^th Edition, 2009.

2.

G.B.Thomas, Calculus, 12^th Edition, Pearson Education India, 2015.

3.

Jain. R.K. and Iyengar. S.R.K., Advanced Engineering Mathematics, Narosa Publications, New Delhi, 5th Edition, 2016.

4.

Peter V.O Neil,  “Advanced Engineering Mathematics”, 7^th Edition, Cengage learning India Pvt Ltd, New Delhi, 2012.

5.

Srimanta Pal, Subodh C Bhunia, “Engineering Mathematics”, Oxford University Press,2015.

COURSE OUTCOMES

At the end of the course students should be able to

CO1

Evaluate gradient, divergence and curl and solve engineering problems involving cubes, rectangular parallelepipeds by applying various integral theorems.  

CO2

Solve the higher order linear differential equations with various methods and apply them in some physical situations.

CO3

Test the analyticity, construct the analytic function and transform complex functions from z-plane to w-plane graphically by using conformal mapping.

CO4

Evaluate real and complex integrals over suitable closed path using various integral theorems.

CO5

Understand the mathematical principles on Laplace transforms and would provide them the ability to formulate and solve some of the physical problems of engineering.