Verify Account

SYLLABUS

UNIT-I

Special Functions : Gamma and Beta Functions – Their properties – Evaluation of improper integrals. Series solutions of ordinary differential equations (Power series and Frobenius method).

UNIT-II

Bessel Functions – Properties – Recurrence relations – Orthogonality. Legendre polynomials – Properties – Rodrigue’s formula – Recurrence relations – Orthogonality.

UNIT-III

Functions of a Complex Variable – Continuity – Differentiability – Analyticity – Properties – Cauchy – Riemann equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions – Milne – Thompson method. Conformal Mapping : Transformation of ez, Inz, z2, sin z, cos z, Bilinear transformation – Translation, rotation, magnification and inversion – Fixed point – Cross ratio – Determination of bilinear transformation.

UNIT-IV

Complex Integration : Line integral – Evaluation along a path and by indefinite integration – Cauchy’s integral theorem – Cauchy’s integral formula – Generalized integral formula. Complex Power Series : Radius of convergence – Expansion in Taylor’s series, Maclaurin’s series and Laurent series. Singular point – Isolated singular point – Pole of order m – Essential singularity.

UNIT-V

Residue – Evaluation of residue by formula and by Laurent series – Residue theorem. Evaluation of integrals of the type. (a) Improper real integrals f(x)dx ∞ −∞ ∫

(b) c+2π c f(cosθ,sinθ)dθ ∫

(c) imxf(x)dx