M.Sc. Mathematics
Semester II

COURSE CODECOURSE NAMECREDITS

JMMH201
Complex Analysis:
UNITI
Functions of complex variables; Limit and continuity, Differentiability; Power Series as an analytic function, Exponential and Trigonometric functions, Complex Logarithms, Zeros of analytic functions.
UNITII
Complex integration, curves in the complex plane , basic properties of complex integrals winding number of a curve; Cauchy–Goursat Theorem, Cauchy’s Integral formula, Morera’s Theorem, Laurent’s series Maximum modulus principle, Schwarz lemma, Liouville’s theorem.
UNITIII
Isolated singularities, removable singularity, poles, Singularity at infinity calculus of residue at finite point, residue at the point at in finite residue theorem, Number of zeros, Poles, Rouche’s Theorem.
UNITIV
Bilinear transformations, their properties and classifications, Definitions and examples of conformal mappings; spaces of analytic functions, Hurwitz’s theorem, Montel’s theorem, Riemann mapping theorem; Mobius transformations.
UNITV
Hypergeometric Series, Generalized Hypergeometric functions, Gamma function and its properties, Riemann Zeta function, Riemann’s functional equation.
Books:
 1. J.B. Conway, Narosa Complex Analysis, Publishing House.
 2. Ruel V. Churchill, Complex Variables and Applications, Tata McGrawHill.
 3. Foundation of Complex Analysis, S. Ponnusamy , Narosa Publishing House.
 4. H.A. Priestly, Introduction to Complex Analysis, Clarendon Press, Oxford.
 5. J.B. Conway, Function of one Complex Variable, SpringerVerlag.
04 
JMMH202
Linear Algebra:
UNITI
Vector Spaces: Definition, General properties of vector spaces; Vector subspaces; Algebra of subspaces;Linear Spans; Row space of Matrix; Linear dependence and independence of vectors; Finitedimensional vector spaces; Dimension of vector space and subspaces; Quotient spaces; Direct sum of spaces;Coordinates; Disjoint subspaces. .
UNITII
Vectors in Rn ,Curves in Rn, Vectors in R3,Vector in C3; Matrices: Addition and scalar multiplication,Transpose of matrix, Square matrices; Systems of linear equations; Diagonalisation; Eigen values and Eigen vectors; Minimal polynomial; CayleyHamilton Theorem; Hermitian & SkewHermitian and unitary matrices; Powers of Matrices; Polynomials in Matrices; Invertible Matrices; Special types of Square Matrices; Complex and Block Matrices.
UNITIII
Linear Transforms; Linear operator; Range and null space of a linear Transformation; Rank and nullity;Product of linear Transformation; Singular Transformation; Representation of linear Transformation by matrix; Dual spaces; Dual Bases; Projections.
UNITIV
Inner Product Spaces: Definition, Euclidean and unitary spaces; Norm and length of vector; Cauchy Schwarz’s inequality and Applications; Orthogonality, Orthogonal Sets and Basis, GramSchmidt orthogonalization process; selfadjoint operators, Complex Inner Product Spaces; Unitary and Normal operators; Projection theorem; Spectral theorem.
UNITV
Bilinear Forms: Definition, Bilinear form as vectors; Matrix of a bilinear form; Symmetric & skew Symmetric bilinear forms.
 1. Vivek Sahai, Vikas Bist; Linear Algebra, Narosa Publishing House.
 2. Sharma & Vashistha, Linear Algebra, Krishna Prakashan Media Ltd.
 3. Schaum’s series Linear Algebra, Tata McGraw Hill.
 4. Kenneth Hoffman & Ray Kunze, Linear Algebra, Pearson Education.
04 
JMMH203
Topology:
UNITI
Metric space, Open sets, closed sets, Convergence, Completeness, Continuity in metric space, Cantor intersection theorem.
UNITII
Topological space, Elementary concept, Basis for a topology, Open and closed sets, Interior and closure of sets, Neighborhood of a point, Limits points, Boundary of a set, Subspace topology , Weak topology, Product topology, Quotient topology.
UNITIII
Continuous maps, Continuity theorems for Open and closed sets, Homeomorphism, Connected spaces, Continuity and connectedness, Components, Totally disconnected space, Locally connected space, Compact space, Limit point compact, Sequentially compact space, Local compactness, Continuity and compactness, Tychonoff theorem.
UNITIV
First and second countable space, T1 spaces, Hausdorff spaces, Regular spaces, Normal spaces, Completely normal space, Completely regular space, Tietz Extention theorem, Metrizability, Uryshon Lemma, Uryshon metrization theorem.
UNITV
Fundamental group function, Homotopy of maps between topological spaces, Homotopy equivalence, Contractible and simple connected spaces, Fundamental groups of S1, and S1x S1 etc., Calculation of fundamental groups of Sn , n>1 using Van Kampen’s theorem , Fundamental groups of a topological group.
Books:
 1. James R. Munkres, Topology, Pearson Education Pvt. Ltd.
 2. J. R Munkres, Topology A First Course, Prentice Hall.
 3. J.L. Kelly, General Topology, Van Nostrand, Reinhold Co.
 4. G.F. Simmons, Introduction to Topology and Mordern Analysis.
 5. K. D. Joshi, Introduction to General Topology, Wiley Eastern Limited.
 6. L. A. Steen and J. A. Seebach Jr, Counterexamples in Topology, H R & Winston.
04 
JMMH204
Mechanics:
UNITI
Moments of inertia, Kineticenergy, Angular momentum.
UNITII
Mechanics of a particle and system ofparticles, kinematics of a rigid body, Euler’s angles.
UNITIII
Euler’s dynamical equations, two dimensional motion of a rigid body, Compound pendulum, Contraints.
UNITIV
D’Alembert’s principle, Lagrange’s equations of motion, Techniques of calculus of variations.
UNITV
Hamilton’s principles, Hamilton’s equations of motion, Contact transformation, Lagrange’s and Poison brackets, Integralin variances, HamiltonJacobi Poisson equations.
Books:
 1. Principle of Mechanics: Singe and Griffith.
 2. Lectures in Analytical Mechanics: F. Gantamacher.
 3. Ele. Treatise on the dynamics of particle and rigid bodies: S.L. Loney.
 4. A Text Book of Dynamics: F. Choelton.
 5. An introduction to the Calculus of Variation: C. Fox.
 6. Calculus of Variations: R. Weinstock.
04 
JMMH205
VivaVoce
04 
Total Credits20