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By Prof S.K.Ray, Department of Mathematics and Statistics IIT Kanpur
UNIT – I
Special functions: Gamma and Beta Functions – Their properties – evaluation of improper integrals. Bessel
functions – properties – Recurrence relations – Orthogonality. Legendre polynomials – Properties – Rodrigue’s
formula – Recurrence relations – Orthogonality.
Functions of a complex variable – Continuity – Differentiability – Analyticity – Properties – Cauchy-Riemann
equations in Cartesian and polar coordinates. Harmonic and conjugate harmonic functions – Milne –
Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties – General power Z
C (c is complex), principal value.
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.
Residue – Evaluation of residue by formula and by Laurent series - Residue theorem.
Evaluation of integrals of the type
(a) Improper real integrals ò¥
f (x)dx (b) ò + p q q q 2 (cos ,sin ) c
eimx f (x)dx (d) Integrals by identation.
Argument principle – Rouche’s theorem – determination of number of zeros of complex polynomials -
Maximum Modulus principle - Fundamental theorem of Algebra, Liouville’s Theorem.
Conformal mapping: Transformation by ez , lnz, z2, z n (n positive integer), Sin z, cos z, z + a/z.
Translation, rotation, inversion and bilinear transformation – fixed point – cross ratio – properties – invariance
of circles and cross ratio – determination of bilinear transformation mapping 3 given points .
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