IIT GUEST LECTURE Mathematics - III VIDEO

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Lecture series on Mathematics-1 by Prof S.K.Ray,
Department of Mathematics and Statistics IIT Kanpur

Syllubas :

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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