What You will Learn ?

Module 1

Partial differential equations, Formation of partial differential equations –elimination of
arbitrary constants-elimination of arbitrary functions, Solutions of a partial differential equations, Equations solvable by direct integration, Linear equations of the first order-
Lagrange’s linear equation, Non-linear equations of the first order -Charpit’s method, Solution of equation by method of separation of variables.

Module 2

One dimensional wave equation- vibrations of a stretched string, derivation, solution of the wave equation using method of separation of variables, D’Alembert’s solution of the wave equation, One dimensional heat equation, derivation, solution of the heat equation

Module 3

Complex function, limit, continuity, derivative, analytic functions, Cauchy-Riemann equations, harmonic functions, finding harmonic conjugate, Conformal mappings - mappings: w = z², w = e^z. Linear fractional transformation w = 1/z, fixed points, Transformation w = sin(z). 

(From sections 17.1, 17.2, and 17.4, only mappings w = z², w = e^z, w = 1/z, w = sin(z), and problems based on these transformations need to be discussed)

Module 4

Complex integration, Line integrals in the complex plane, Basic properties, First evaluation
method-indefinite integration and substitution of limit, second evaluation method-use of a
representation of a path, Contour integrals, Cauchy integral theorem (without proof) on
simply connected domain,Cauchy integral theorem (without proof) on multiply connected
domain Cauchy Integral formula (without proof), Cauchy Integral formula for derivatives of
an analytic function, Taylor’s series and Maclaurin series.,

Module 5

Laurent’s series (without proof), zeros of analytic functions, singularities, poles, removable singularities, essential singularities, Residues, Cauchy Residue theorem (without proof), Evaluation of definite integral using residue theorem, Residue integration of real integrals – integrals of rational functions of cos(θ) and sin(θ), integrals of improper integrals of the form ∫[−∞ to ∞] f(x) dx with no poles on the real axis. (∫[A to B] f(x) dx whose integrand becomes infinite at a point in the interval of integration is excluded from the syllabus),