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 VARIABLEIntroduction to complex numbersThe real and imaginary parts of a complex numberModulus and Argument of a complex numberComplex number as vectorsCartesian and Polar representation of complex numbersPure Real and Imaginary numbersRepresentation of complex numbers on a Riemann SphereAlgebraic operations...
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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 VARIABLEIntroduction to complex numbersThe real and imaginary parts of a complex numberModulus and Argument of a complex numberComplex number as vectorsCartesian and Polar representation of complex numbersPure Real and Imaginary numbersRepresentation of complex numbers on a Riemann SphereAlgebraic operations on complex numbers.Conformal MappingInverse TransformationSeries of complex numbersCauchy's TheoremCauchy's Theorem of ResidueCauchy's Integral formula10. THEORY OF PROBABILITYMathematical definition of probabilityExclusive eventsIndependent eventsRules governing probability of independent eventsDependent eventsBinomial law of probabilityRandom variablesDiscrete distributionContinuous distributionJoint distribution of several variablesExpectationParameters of a frequency distributionSheppard's correctionBinomial frequency distributionStirling's formula for the approximation of factorials of large numbersChekychef's TheoremNormal frequency distribution or Gaussian distributionPoission's distributionLaw of large numbersCorrelationScatter diagramStandard error of the mean SEM11. LAPLACE TRANSFORMSThe Laplace transformation.The Laplace transformation of the sum of two functionsSectionally or piecewise continuous functionsFunctions of exponential orderInverse Laplace transformsTransforms of derivativesTransforms of integralsThe first shift theorem of multiplying the object function by eatLaplace's solution of linear differential equations with constant coefficientsThe second translation or shifting propertyThe unit impulse functionThe unit doubletDifferentiation of transformThe Heaviside's expansion formulaeUsing Laplace Transformation in solving Linear Partial Differential Equations12. ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONSTotal Differential EquationsOrdinary differential equationsLinear simultaneous equations with constant coefficientsSimultaneous Total Differential EquationsFirst Order differential equations Separable Equations Homogeneous Equations Differential equations of the second orderLinear differential equations of the second orderPartial Differential Equations of the First OrderNon linear partial differential equations of the first orderDamped mechanical oscillationsElectrical oscillations- RCL circuit13. APPLICATIONS IN ENGINEERING AND PHYSICS14. SOLVED PROBLEMS AND EXERCISES