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UNIT-I FUNCTIONS OF A COMPLEX VARIABLE

1.1 Introduction – Continuity – Differentiability-Analyticity – Properties

1.2 Cauchy-Riemann equations in Cartesian and polar coordinates

1.3 Harmonic and conjugate harmonic functions – Milne – Thompson method

UNIT-II INTEGRATION AND SERIES EXPANSIONS

2.1 Complex integration: Line integral – Cauchy’s integral theorem , Cauchy’s integral formula

2.2 Generalized integral formula (all without proofs) - Radius of convergence

2.3 Expansion in Taylor’s series, Maclaurin’s series and Laurent series

UNIT III INTEGRATION USING RESIDUES

3.1 Types of Singularities: Isolated, pole of order m, essential

3.2 Residues

3.3 Residue theorem( without proof)

3.4 Evaluation of real integrals of type (a) (b) (c)

UNIT IV CONFORMAL MAPPING

4.1 Transformation by exp z, lnz, z^2, z^n(n positive integer), Sin z, cos z, z + a/z-

4.2 Translation, rotation, inversion and bilinear transformation – fixed point – cross ratio – properties – invariance of circles.

UNIT V SAMPLING DISTRIBUTIONS

5.1 Review of Normal distribution - Population and samples

5.2 Sampling distribution of mean (with known and unknown variance)Proportion, variances

5.3 Sampling distribution of sums and differences -Point and interval estimators for means, variances, proportions

UNIT VI TESTS OF HYPOTHESIS

6.1 Type I and Type II errors

6.2 Maximum error- One tail, two-tail tests - Tests concerning one mean and proportion, two means

6.3 Proportions and their differences using Z-test, Student’s t-test - F-test and Chi -square test.