**Jntu Mathematics 3 syllabus**

**MATHEMATICS � III**

**UNIT � I:**Special functions I:

Review of Taylor�s series for a real many valued functions, Series solutions to Differential Equations. Gamma and Beta Functions � Their properties � evaluation of improper integrals. Bessel functions � properties � Recurrence relations � Orthogonality.

**UNIT-II**Special functions II:

Legendre polynomials � Properties � Rodrigue�s formula � Recurrence relations � Orthogonality. Chebycher�s polynomials - properties � Recurrence relations � Orthogonality.

**UNIT-III:**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.

Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties � General power Z (c is complex), principal value Logarithmic function.

**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.( distribution between the real analyticity and complex analyticity )

**UNIT-VI:**Contour integration:

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 :**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 .

**UNIT-VIII:**Elementary Graph Theory:

Graphs, representation by matrices adjacent matrix-incident matrix- simple, multiple, regular, complete, Bipartite & Planar graphs-

Hamiltonian and Eulerian Circuits- �Trees� Spanning tree minimum spanning trees.

**Text Books:**

1. Engineering Mathematics, III by P B Bhaskara rao, SK VS RAMACHARY M Bhujanaga rao & othrs

2. A text Book of Engineering Mathematics, C. Sankaraiah, V. G. S. Book Links.

**References:**

1. A text Book of Engineering Mathematics, Vol-III T. K. V. Iyengar, B. Krishna Gandhi and Others, S. Chand & Company.

2. Higher engineering mathematics by b.s.grewal khanna pub

3. Adv. Eng. Maths by jain & srk iyengar, narsa publications.

4. Complex varaiables by rv Churchill

5. Adv. Eng. Maths by allen Jeffery academic press.