Detailed Syllabus

• (a) Complex Analysis
  • Field of Complex Numbers; Why the field of complex numbers cannot be ordered; Absolute value of a complex number and its properties; Conjugate of a complex number and its properties; Polar form of a complex number; Argument of a complex number; Demoivre’s theorem, nth root of a complex number; geometrical interpretation of all these concepts; Use of these ideas in trigonometry.
  • Complex valued functions; Concepts of limits, continuity, differentiability, and analyticity of a complex function; Cauchy-Riemann Equations and its use; harmonic functions; conjugate harmonic functions.
  • Elementary complex functions: exponential function and its periodicity; Logarithmic function as the inverse of exponential function; Trigonometric/hyperbolic functions in terms of exponential function; Inverse trigonometric functions; complex powers of a complex variable; principle values of multi-valued functions.
  • Conformality of analytic functions; mappings of various plane regions by means of elementary analytic functions.
  • Mobius transformations; composition of Mobius transformations; Mobius transformation as a composition of translation, dilatation, rotation, inversion; fixed points of a Mobius transformation; Invariance of cross-ratios through Mobius transformation; mapping of disks/half planes using Mobius transformations; determination of Mobius transformation which map a given region onto another prescribed region.
  • Complex Integration: Line integral and its properties; ML- inequality; dependence of integral along a path upon the end points of the path; Caughy’s Integral Theorem for a simply connected domain and its extension to multiply connected domains; Existence of anti-derivative.
  • Cauchy’s Integral Formula; existence of derivatives of all orders of an analytic function; Cauchy’s Inequality; Liouville’s Theorem; Morera’s Theorem; Fundamental Theorem of Algebra; Gauss’s Theorem (zeros of the derivative of a polynomial lie within the convex hull of the zeros of the polynomial).
  • Sequences and Series of complex numbers; Convergence, Absolute convergence, Conditional convergence of a series of complex numbers; Tests of convergence of a series (Cauchy convergence criterion, comparison test, ratio test, root test, Leibnitz test). Power Series; radius of convergence of a power series, Identity theorm for power series; Power series represents an analytic function inside the circle of convergence.
  • Taylor’s Series of an analytic function; determination of the radius of convergence of a Taylors’s series; Laurent’s series; methods of obtaining Laurent’s series.
  • Zeros of an analytic function are isolated; form of an entire function which has no zeros in the whole complex plane; Isolated/non-isolated singularities; classification of isolated singularities as removable, pole, essential.; Residue of a complex function at an isolated singularity; Cauchy’s Residue theorem for evaluation of a contour integral; Rouche’s Theorem.
  • Evaluation of some real integrals (proper/improper) by using the Cauchy’s Residue theorem.
• (a) Matrices
  1. Matrices as arrays of numbers; equality of two matrices, addition of two matrices of the same order; multiplication of a matrix by a scalar; multiplication of two matrices; properties of these basic matrix operations.
  2. Identity matrix, Diagonal matrices; Scalar matrices; Symmetric/Skew-symmetric matrices; Hermitian/Skew-Hermitian matrices; Orthogonal/Unitary matrices; Idempotent and Involutory matrices; Nilpotent matrices. (These special matrices were used to show that some results are not true for arbitrary matrices, but the same result may be valid when restricted to a suitable class of matrices. Even otherwise, such matrices play an important role when more concepts are introduced)
  3. Transpose of a matrix and its properties; Trace of a square matrix and its properties; Inverse of a matrix and its properties.
  4. Polynomial function of a matrix; Polynomial matrix equationp(A) = 0 has an infinite number of solutions. (Polynomial functions of a matrix play an important role later on. At this stage, this was introduced to give meaningful practice in matrix operations).
  5. Vector spaces, Sub-spaces; Linear combinations; Span of a set of vectors; Linear Dependence/Linear Independence; Basis of a vector space; Finite dimensional vector spaces; Dimension of a finite dimensional vector space; Extension of a linearly independent set to a basis of the vector space; Reduction of a spanning set to a basis of the vector space; Co-ordinates of a vector with respect to a basis; Sum/Intersection of sub-spaces; Complementary sub-spaces; Dimension of the sum of two sub-spaces.
  6. Inner Products; Norm of a vector; Cauchy-Schwarz Inequality; Triangle Inequality; Orthonormal basis; Gram-Schmidt orthogonalisation process.
  7. Row/Column space of a matrix; Row/Column rank of a matrix; Rank of a matrix; Rank one matrices; Null space of a matrix; Rank-Nullity theorem; Geometrical interpretation of the solution set of a system of m linear equations in n unknowns.
  8. Row-reduced Echelon form of a matrix; Use of the row-reduced echelon form of a matrix to compute the rank of a matrix, solutions of the matrix equation AX = b, and the inverse of a matrix.
  9. Partitioning of a matrix; Computing the inverse of a symmetric matrix recursively; Using the inverse of a symmetric matrix to compute the inverse of an arbitrary non-singular matrix.
  10. Eigen values/Eigen vectors of a matrix; Characteristic polynomial; Diagonalisable matrices; Eigen values of a symmetric/Hermitian matrix are real; Orthogonality of the eigen vectors of a symmetric/Hermitian matrix corresponding to distinct eigen values; .Similarity of matrices.

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• Syllabus