MATHEMATICS – III

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.

UNIT-II
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. 

UNIT-III
Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties – General power Z (c is complex), principal value.

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. 

UNIT-V
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-VI
Residue – Evaluation of residue by formula and by Laurent series - Residue theorem.
Evaluation of integrals of the type 
(a) Improper real integrals (b) 
(c) (d) Integrals by identation.

UNIT-VII
Argument principle – Rouche’s theorem – determination of number of zeros of complex polynomials - Maximum Modulus principle - Fundamental theorem of Algebra, Liouville’s Theorem.

UNIT-VIII
Conformal mapping: Transformation by , lnz, z2, z (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 .

Text Books:
1.A text Book of Engineering Mathematics, Vol-III T. K. V. Iyengar, B. Krishna Gandhi and Others, S. Chand & Company.
2.A text Book of Engineering Mathematics, C. Sankaraiah, V. G. S. Book Links.
3.A text Book of Engineering Mathematics, Shahnaz Bathul, Prentice Hall of India.
4.A text Book of Engineering Mathematics, P. Nageshwara Rao, Y. Narasimhulu & N. Prabhakar Rao, Deepthi Publications.

References:
1.A text Book of Engineering Mathematics, B. V. Raman, Tata Mc Graw Hill.
2.Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd.
3.A text Book of Engineering Mathematics, Thamson Book Collection.