Subject Code:53007 L:3 T/P/D:1 Credits:3 Int. Marks:25 Ext. Marks:75 Total Marks:100
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: 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: III
Elementary functions: Exponential, trigonometric, hyperbolic functions and their properties – General power Z C (c is complex), principal value.
UNIT IV: 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: 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: VI
Residue – Evaluation of residue by formula and by Laurent series - Residue theorem. Evaluation of integrals of the type ∞ (a) Improper real integrals ∫−∞ f ( x)dx c +2π (b) ∫c f (cosθ , sin θ )dθ ∞ (c) ∫−∞ e imx f ( x)dx (d) Integrals by identation.
UNIT VII: 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: VIII
Conformal mapping: Transformation by e z , 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.
REFERENCE BOOKS:
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.