Probability – Probability models and axioms, conditioning and Bayes’ rule Discrete random variables; probability mass functions; expectations, examples, multiple discrete random variables: joint PMFs, expectations, conditioning, independence Continuous random variables, probability density functions, expectations, examples, multiple continuous random variables, continuous Bayes rule, covariance and correlation. Statistics – Bayesian statistical inference, point estimators, parameter estimators, test of hypotheses, tests of significance.
Complex numbers, complex plane, polar form of complex numbers. Powers and roots, derivative. Analytic functions, Cauchy Riemann equations, Laplace equation, conformal mapping. Exponential function, trigonometric functions, hyperbolic functions, logarithms, general power and linear fractional transformation.
Complex line integral, Cauchy integral theorem, Cauchy integral formula, derivatives of analytical functions. Power series, Taylor series and McLaurin series. Laurent series, zeroes and singularities, residues, Cauchy residue theorem, evaluation of real integrals using residue theorem.