PROBABILITY AND COMPLEX FUNCTIONS

520 364

First Edition : August 2022

UNIT I : PROBABILITY AND RANDOM VARIABLES 

Axioms of probability – Conditional probability – Baye’s theorem – Discrete and continuous random
variables – Moments – Moment generating functions – Binomial, Poisson, Geometric, Uniform,
Exponential and Normal distributions – Functions of a random variable.

UNIT II : TWO-DIMENSIONAL RANDOM VARIABLES 

Joint distributions – Marginal and conditional distributions – Covariance – Correlation and linear
regression – Transformation of random variables – Central limit theorem (for independent and
identically distributed random variables).

UNIT III : ANALYTIC FUNCTIONS 

Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar
coordinates – Properties – Harmonic conjugates – Construction of analytic function – Conformal
mapping – Mapping by functions  w=z+c, cz 1/z ,z2
– Bilinear transformation.

UNIT IV :  COMPLEX INTEGRATION 

Line integral – Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series
– Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real
integrals – Applications of circular contour and semicircular contour (with poles NOT on real axis).

UNIT V : ORDINARY DIFFERENTIAL EQUATIONS 

Higher order linear differential equations with constant coefficients – Method of variation of parameters
– Homogenous equation of Euler’s and Legendre’s type – System of simultaneous linear first order
differential equations with constant coefficients – Method of undetermined coefficients.

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