**Download/Source url :**

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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