Educational videos of Mathematics - III

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http://www.youtube.com/watch?v=-xvTzwot9MI&feature=player_embedded

By Prof S.K.Ray, Department of Mathematics and Statistics IIT Kanpur

MATHEMATICS – III SYLLABUS :

Source :http://jntuk.edu.in

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 (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 ò¥
-¥
f (x)dx (b) ò + p q q q 2 (cos ,sin ) c
c
f d
(c) ò¥
-¥
eimx f (x)dx (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 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 .


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.

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