This is Part II of my personal study notes in advanced mathematics. It entails theoretical analysis, solved examples, and exercises on the following topics:
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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
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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
