Department of Mathematics

Understand the concepts of different types of graphs.

Text: John B. Conway, Functions of One Variable, Second

Edition.

Module 1: Analytic functions, Power series, Analytic functions as mappings, Mobius

Transformations.(Chapter 3 of the text)

(20 hours)

Module 2: Power series representation of analytic functions, Zeros of an analytic function. The index

of a closed curve, Cauchy’s theorem and Cauchy’s integral formula, The Homotopic version of

Cauchy’s theorem and simple connectivity

(Chapter 4 – Sections 4.2 – 4.6 of the text. ) (20 hours.)

Module 3: Counting zeros, The open mapping theorem, Goursat’s Theorem, Classification of

singularities, Residues, The Argument Principle.

(Chapter 4 – Sections 7 and 8 of the text) (25 hours.)

Module 4: The Maximum Modulus theorem, The maximum principle, Schwarz’s lemma, Convex

functions and Hadmard’s Three Circles Theorem.

(Chapter 6 – Sections 1 - 3 of the text) (25 hours.)

Course  Objectives: 

The  objectives  of  the  course  include  teaching  the  students  the  concepts  of  Fourier  Series,  Fourier and  Laplace  Transforms  and  their  applications  in  the  physical  world. The  course  also  introduces  the concept  of groups  which  is  very useful  in  studying symmetry  of molecular  structures.

Syllubus

Module 1                                                                                                          (23 hrs)

Fourier SeriesPeriodic functions, Fourier series, Euler’s formulae, Dirichlet’s conditions, Change of interval, Halfrange series

Module  II                                                                                                       (23 hrs)

 Laplace  Transforms Definitions,   Properties,   Inverse   Laplace   transforms,   Convolution   theorem,   Application   to differential  equations.

COURSE: 16P2MATT08: ADVANCED COMPLEX ANALYSIS Hours per week: 5 Total Credits: 4

 Text Book: Lars V. Ahlfors, Complex Analysis, Third edition, McGraw Hill Internationals 

Module 1: Elementary theory of power series: sequences, series, uniform convergence, power series, Abel’s limit theorem. Power series expansions: Weierstrass’ theorem, the Taylor’s series, the Laurent’s series Partial fractions and factorization: partial fractions, infinite products, canonical products, the gamma functions. (Chapter 2, Section 2 - Chapter 5, Sections 1, 2.1 to 2.4 of the text) (20 hours) 

Module 2: Entire functions: Jenson’s formula, Hadamard’s theorem (without proof) the Riemann zeta function: the product development, extension of ξ to the whole plane, the functional equation, the zeroes of zeta function Arzela’s theorem (without proof) (Chapter 5 - Sections 3, 4, and 5.3 of the text) (20 hours) 

Module 3: The Riemann mapping theorem: statement and proof, boundary behavior, use of reflection principle, analytic arcs. Conformal mappings of polygons: the behavior of an angle, the Schwarz Christoffel formula (Statement only). A closer look at harmonic functions: functions with mean value property, Harnack’s principle. The Dirichlet problem: sub harmonic functions, solution of Dirichlet problem (statement only) (Chapter 6 Section 1, 2.1, 2.2, 3, 4.1 & 4.2 of the text) (20 hours) 

Module 4: Elliptic functions: simply periodic functions, representation of exponentials, the Fourier development, functions of finite order. Doubly periodic functions: The period module, unimodular transformations, the canonical basis, general properties of elliptic functions. The Weirstrass theory: the Weierstrass function, the functions ξ (y) and σ (y), the differential equation. (Chapter 7 Sections 1, 2, 3 of the text) (15 hours)

In this session we study the Euclidean distance function for the plane has three properties: non-negativity, symmetry and the Triangle Inequality. In a similar way, we find in this extract that the Euclidean distance on Rⁿ satisfies the same three properties. The creative leap is to use this observation to raise these properties to the status of axioms and to say that any function that satisfies them is a well-defined distance function, known as a metric. The extract is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of pure mathematics.

This course analyze the basic techniques for the efficient numerical solution of problems in science and engineering. Topics spanned root finding, interpolation, approximation of functions, differential equations, direct and iterative methods.

CO1: Analyze analytic functions, Power series and Mobius Transformations. 

CO2: Determine power series for analytic functions and its zeros. The index of a closed curve, Cauchy’s theorem and Cauchy’s integral formula, The Homotopic version of Cauchy’s theorem and simple connectivity. 

CO3: Interpret counting zeros, the open mapping theorem, Goursat’s Theorem, classification of singularities, residues and the Argument Principle. 

CO4: Understand Maximum Modulus theorem, maximum principle, Schwarz’s lemma, convex functions and Hadmard’s Three Circles Theorem.

Module 1: TheWeirstrasstheorem,otherformsofFourierseries,theFourierintegraltheorem,theexponentialformofth e Fourierintegraltheorem,integraltransformsandconvolutions,theconvolutiontheoremforFouriertransfor ms. (Chapter 11 Sections 11.15 to 11.21 ofText1) (16 hours)

 Module 2: Multivariable Differential Calculus The directional derivative, directional derivatives and continuity, the total derivative, the total derivative expressed in terms of partial derivatives, An application of complex- valued functions, the matrix of a linear function, the Jacobian matrix, the chain rate matrix form of the chain rule. (Chapter 12 Sections.12.1to12.10ofText1) (17hours.)

Module 3: Implicit functions and extremum problems, the mean value theorem for differentiable functions, a sufficient condition for differentiability, a sufficient condition for equality of mixed partial 47 derivatives, functions with non-zero Jacobian determinant, the inverse function theorem (without proof), the implicit function theorem (without proof), extrema of real- valued functions of one variable, extrema of real- valued functions of several variables. Chapter 12 Sections-. 12.11 to 12.13. of Text 1 Chapter 13 Sections-. 13.1 to 13.6 of Text 1 

Module 4: Integration of Differential Forms Integration, primitive mappings, partitions of unity, change of variables, differential forms, Stokes theorem (without proof) Chapter 10 Sections. 10.1 to 10.25, 10.33 ofText2 (21 hours.)