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Important questions
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2.Countour integration **
3.Laurent series
UNIT-5
1.Convolution theorem (8)
2.f(t)=?(8)
1.CMethod of variation of parameters**
2.Homogenous equation of Euler’s **and Legendre’s type
*These questions are expected for the exams This may or may not be asked for exams .
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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 1 2 z z w z c, cz, , - 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
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 1 2 z z w z c, cz, , - 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