ANDHRA UNIVERSITY, WALTAIR
M.Sc. APPLIED MATHEMATICS , FIRST SEMESTER
AM-102 : COMPLEX ANALYSIS
[ A total of seven questions are to be set. All questions carry equal marks. The first question which is compulsory carries 20 marks. It consists of 4 short answer sub questions each of five marks covering the entire syllabus. The remaining six questions each carrying 20 marks are to be set as suggested in the body of the syllabi.]
Linear equations with variable coefficients , the Wronskian and linear independence, reduction of the order of a homogeneous equations, the non-homogeneous equation. Homogeneous equations with analytic coefficients , Linear equations with regular singular points, Eulers equations series solutions. Existence and uniqueness of solutions of 1st order equations, exact equations, Picard’s method of successive approximation,s existence & uniqueness of solution to systems. (Chapter 3 (excluding section 8 &9), chapter 4 (excluding sections 5,7 &8), chapter 5 (only sections 1,3,5,6) of Ref. 1).
(Three questions are to be set)
Calculus of variations : Euler’s equations , functions of the form 
Functional
dependent on higher order derivatives, variational problems in parametric form
and applications (chapter VI of Ref.2).
Tensor Analysis : N-dimensional space, covariant and contravariant vectors, contraction, second & higher order tensors, quotient law, fundamental tensor, associate tensor, angle between the vectors, principal directions, christoffel symbols, covariant and intrinsic derivatives geodesics (chapter 1 to 4 of Ref.3).
(Three questions are to be set)
______________________________________________________________________________________________________
References:
- E.A.Coddington : An introduction to ordinary differential equations, prentice hall of India, Pvt., Ltd., New Delhi, 1987.
- L.Elsgolts : Differential equations and calculus of variations, Mir publishers, Moscow, 1971.
- Barry spain : Tensor calculus, Radha publishing house, Calcutta, 3rd edition.
