• Text and references

  Lecture materials are taken from the following books:

  1 - Engineering Mathematics (in Farsi), J. Rashed Mohassel, 1393, University of Tehran Publications

  2 - Complex Variables and Applications, 8^th Ed., J. W. Brown and R. V. Churchill, 2009, McGraw-Hill Inc. (Chapters 1 to 9)

  3 - Fourier Series and Boundary Value Problems, 5^th Ed., J. W. Brown and R. V. Churchill, 1993, McGraw-Hill Inc. (Chapters 1 to 6)

  4 - Advanced Engineering Mathematics, 10^th Ed., E. Kreyszig, 2011, John Wiley & Sons Inc. (Chapters 11 to 17)

  5 - Advanced Engineering Mathematics, 6^th Ed., C. R. Wylie and L. C. Barrett, 1995, McGraw-Hill Inc. (Chapters 8, 9, 11, 17 to 20)

  6 - Advanced Engineering Mathematics, A. Jeffrey, 2002, Academic Press (Chapters 9, 10, 13 to 18)

• Grading policy

Mid-term exam: 8 - 10

Final exam: 8 - 10

Homework, quizzes, attendance: 2 - 3

Quizzes: There will be 4 quizzes during the term taken in tutorial classes.

Attendance: If you are absent in 6 lectures, you fail the course with no exceptions. Students are not allowed in class 15 minutes after the lecture starts

• TENTATIVE course outline:

Fourier Analysis

à    Fourier series (real and complex forms), approximation of periodic functions

à    Even and odd functions, half range expansions

à        Convergence of Fourier series, absolute and uniform convergence, Gibbs phenomenon, integration and differentiation of Fourier series

à        Bessel inequality, Parseval theorem

à        (*) Double Fourier series

à        Fourier transform, Sine and Cosine transforms, relation with Fourier series, Fourier transform properties

Partial Differential Equations

à    Basic concepts of PDEs, classification of PDEs

à    Separation of variables, simple illustrative examples

à    Heat conduction problems in one dimension, two point boundary value problems

à    Sturm-Liouville problems, orthogonality of eigenfunctions, generalized Fourier series

à    Heat equation, wave equation, Laplace equation in rectangular and cylindrical coordinates

à    Solution in finite and semi-infinite domains, applications of Fourier and Laplace transforms

à        (*) Laplace equation in spherical coordinates

 

Mid-term Exam: Thu 95/02/09 8:30AM

 

Complex Analysis

à    Complex numbers, geometrical interpretation, polar representation

à    Products, powers, and roots of complex variables

à    Functions of a complex variable, continuity, limit, and derivative, analytic functions

à    Cauchy-Riemann equations, harmonic functions

à    Mappings by elementary complex functions: exponential function, trigonometric functions, hyperbolic functions, logarithmic function

à    Multi-valued functions, branch points, branch cuts, Riemann sheets

à    Contour integrals in complex plane, Cauchy-Goursat theorem, Cauchy integral formula,

derivatives of analytic functions, Liouville theorem, Morera's theorem, maximum modulus theorem

à    Infinite sequences and series, power series, Taylor series, Laurent series, uniform and absolute convergence of power series

à    Singular points of complex functions, classification of singular points, residues, Cauchy residue theorem, applications of residue theorem in evaluation of proper and improper definite integrals, Jordan's lemma

à    (*) Inverse Laplace transform

à    (*) Argument principle, zeros of analytic functions

à    Conformal mapping and its applications in solving the Laplace equation