Advanced Mathematics Personal Study Notes- Part II

This is Part II of my personal study notes in advanced mathematics. It entails theoretical analysis, solved examples, and exercises on the following topics: 9. FUNCTIONS OF COMPLEX VARIABLE Introduction to complex numbers The real and imaginary parts of a complex number Modulus and Argument of a complex number Complex number as vectors Cartesian and Polar representation of complex numbers Pure Real and Imaginary numbers Representation of complex numbers on a Riemann Sphere Algebraic operations on complex numbers. Conformal Mapping Inverse Transformation Series of complex numbers Cauchy's Theorem Cauchy's Theorem of Residue Cauchy's Integral formula 10. THEORY OF PROBABILITY Mathematical definition of probability Exclusive events Independent events Rules governing probability of independent events Dependent events Binomial law of probability Random variables Discrete distribution Continuous distribution Joint distribution of several variables Expectation Parameters of a frequency distribution Sheppard's correction Binomial frequency distribution Stirling's formula for the approximation of factorials of large numbers Chekychef's Theorem Normal frequency distribution or Gaussian distribution Poission's distribution Law of large numbers Correlation Scatter diagram Standard error of the mean SEM 11. LAPLACE TRANSFORMS The Laplace transformation. The Laplace transformation of the sum of two functions Sectionally or piecewise continuous functions Functions of exponential order Inverse Laplace transforms Transforms of derivatives Transforms of integrals The first shift theorem of multiplying the object function by eat Laplace's solution of linear differential equations with constant coefficients The second translation or shifting property The unit impulse function The unit doublet Differentiation of transform The Heaviside's expansion formulae Using Laplace Transformation in solving Linear Partial Differential Equations 12. ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS Total Differential Equations Ordinary differential equations Linear simultaneous equations with constant coefficients Simultaneous Total Differential Equations First Order differential equations Separable Equations Homogeneous Equations Differential equations of the second order Linear differential equations of the second order Partial Differential Equations of the First Order Non linear partial differential equations of the first order Damped mechanical oscillations Electrical oscillations- RCL circuit 13. APPLICATIONS IN ENGINEERING AND PHYSICS 14. SOLVED PROBLEMS AND EXERCISES