Course Objectives:
To familiarize the complex variables.
To make the student capable of evaluating the integrals in complex domains
To make the student capable of expanding a given function as a series and finding the poles
and residues
To make the student capable of evaluating the integrals in complex domains using residue
theorem
To familiarize the students with the foundations of probability and statistical methods.
To equip the students to solve application problems in their disciplines.
Course Outcomes: At the end of the course students will be able to
● apply Cauchy-Riemann equations to complex functions in order to determine whether a given
continuous function is analytic (L3)
● find the differentiation and integration of complex functions used in engineering problems
(L5)
● make use of the Cauchy residue theorem to evaluate certain integrals (L3)
● apply discrete and continuous probability distributions (L3)
● design the components of a classical hypothesis test (L6)
● infer the statistical inferential methods based on small and large sampling tests (L4)
UNIT – I: Functions of a complex variable and Complex integration:
Introduction – Continuity – Differentiability – Analyticity –Cauchy-Riemann equations in Cartesian
and polar coordinates – Harmonicand conjugate harmonic functions – Milne – Thompson method.
Complex integration: Line integral – Cauchy’s integral theorem – Cauchy’s integral formula –
Generalized integral formula (all without proofs) and problems on above theorems.
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UNIT – II:Series expansions and Residue Theorem:
Radius of convergence – Expansion in Taylor’s series, Maclaurin’s series andLaurent series.
Types of Singularities: Isolated – Essential –Pole of order m– Residues – Residue theorem
(without proof) – Evaluation of real integral of the type
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UNIT – III: Probability and Distributions:
Review of probability and Baye’s theorem – Random variables – Discrete and Continuous random
variables – Distribution functions – Probability mass function, Probability density function and
Cumulative distribution functions – Mathematical Expectation and Variance – Binomial, Poisson,
Uniform and Normal distributions.
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UNIT – IV: Sampling Theory:
Introduction – Population and Samples – Sampling distribution of Means and Variance (definition
only) – Central limit theorem (without proof) – Representation of the normal theory distributions –
Introduction to t, and F-distributions – Point and Interval estimations – Maximum error of estimate.
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UNIT – V: Tests of Hypothesis:
Introduction – Hypothesis – Null and Alternative Hypothesis – Type I and Type II errors – Level of
significance – One tail and two-tail tests – Tests concerning one mean and two means (Large and
Small samples) – Tests on proportions.
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Text Books:
1. B. S. Grewal, Higher Engineering Mathematics, 44th Edition, Khanna Publishers.
2. Miller and Freund’s, Probability and Statistics for Engineers,7/e, Pearson, 2008.
Reference Books:
1. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 9th edition, Mc-
Graw Hill, 2013.
2. S.C. Gupta and V.K. Kapoor, Fundamentals of Mathematical Statistics, 11/e, Sultan Chand
& Sons Publications, 2012.
3. Jay l. Devore, Probability and Statistics for Engineering and the Sciences, 8th
Edition,Cengage.
4. Shron L.Myers, Keying Ye, Ronald E Walpole, Probability and Statistics Engineers and the
Scientists,8th Edition, Pearson 2007.
5. Sheldon, M. Ross, Introduction to probability and statistics Engineers and the Scientists,
4thEdition, Academic Foundation,2011