Complex Analysis & Calculus of Variations

SYLLABUS- COMPLEX ANALYSIS & CALCULUS OF VARIATIONS, Section A
Complex Analysis
Analytic function. Cauchy-Riemann equations. Harmonic functions. Complex
integration. Cauchy's theorem. Cauchy's integral formula. Derivatives. Taylor's series.
Laurent's series. Liouville's theorem. Morera's theorem. Zeros and singularities. Poles
and residues. Cauchy's residue theorem. Contour integration. Rouche's theorem. Hurwitz theorem. Jensen's theorem. (4 questions)
Section B
Expansion of simple functions in Fourier series. Fourier transform and its simple
properties. (2 questions)
Calculus of Variations
Functionals. Variation of a functional. Euler's equation. Case of several variables.
Natural boundary conditions. Variational derivative. Invariance of Eular's equation
under transformation of coordinates. fixed end point problem for n unknown
functions. Variational problem with subsidiary conditions.
Isoperimetric problem. Finite subsidiary conditions. (2 questions)