Engineering Mathematics II (Semester III) for UPTU

Cover
Contents
Preface
Chapter 1: Functions of Complex Variables
	1.1 Basic Concepts
	1.2 Analytic Functions
	1.3 Integration of Complex-Valued Functions
	1.4 Power Series Representation of an Analytic Function
	1.5 Zeros and Poles
	1.6 Residues and Cauchy’s Residue Theorem
	1.7 Evaluation of Real Definite Integrals
	1.8 Conformal Mapping
	1.9 Miscellaneous Examples
	Exercises
Chapter 2: Elements of Statistics and Probability
	2.1 Measures of Central Tendency
	2.2 Measures of Variability (Dispersion)
	2.3 Measures of Skewness
	2.4 Measures of Kurtosis
	2.5 Curve Fitting
		Least Square Line Approximation
		The Power Fit y = axm
		Least Square Parabola (Parabola of Best Fit)
	2.6 Covariance
	2.7 Correlation and Coefficient of Correlation
	2.8 Regression
	2.9 Angle Between the Regression Lines
	2.10 Probability
	2.11 Conditional Probability
	2.12 Independent Events
	2.13 Probability Distribution
	2.14 Mean and Variance of a Random Variable
	2.15 Binomial Distribution
	2.16 Pearson’s Constants for Binomial Distribution
	2.17 Poisson Distribution
	2.18 Constants of the Poisson Distribution
	2.19 Normal Distribution
	2.20 Characteristics of the Normal Distribution
	2.21 Normal Probability Integral
	2.22 Areas Under the Standard Normal Curve
	2.23 Fitting of Normal Distribution to a Given Data
	2.24 Sampling
	2.25 Level of Significance and Critical Region
	2.26 Test of Significance for Large Samples
	2.27 Confidence Interval for the Mean
	2.28 Test of Significance for Single Proportion
	2.29 Test of Significance for Difference of Proportion
	2.30 Test of Significance for Difference of Means
	2.31 Test of Significance for the Difference of Standard Deviations
	2.32 Sampling with Small Samples
	2.33 Significance Test of Difference Between Sample Means
	2.34 Chi-Square Distribution
	2.35 X2-Test as a Test of Goodness-of-Fit
	2.36 Snedecor’s F-Distribution
	2.37 Fisher’s Z-Distribution
	2.38 Analysis of Variance (Anova)
	2.39 Forecasting and Time Series Analysis
	2.40 Statistical Quality Control
	2.41 Miscellaneous Examples
	Exercises
Chapter 3: Non-Linear Equations
	3.1 Classification of Methods
	3.2 Approximate Values of the Roots
	3.3 Bisection Method (Bolzano Method)
	3.4 Regula–Falsi Method
	3.5 Convergence of Regula–Falsi Method
	3.6 Newton–Raphson Method
	3.7 Square Root of a Number Using Newton–Raphson Method
	3.8 Order of Convergence of Newton–Raphson Method
	3.9 Fixed Point Iteration
	3.10 Convergence of Iteration Method
	3.11 Square Root of a Number Using Iteration Method
	3.12 Sufficient Condition for the Convergence of Newton–Raphson Method
	3.13 Newton’s Method for Finding Multiple Roots
	3.14 Newton–Raphson Method for Simultaneous Equations
	Exercises
Chapter 4: Linear Systems of Equations
	4.1 Direct Methods
		Matrix Inversion Method
		Gauss Elimination Method
		Jordan Modification to Gauss Method
		Triangularization (Triangular Factorization) Method
		Triangularization of Symmetric Matrix
		Crout’s Method
	4.2 Iterative Methods for Linear Systems
		Jacobi Iteration Method
		Gauss–Seidel Method
		Convergence of Iteration Method
	4.3 ILL-Conditioned System of Equations
	Exercises
Chapter 5: Finite Differences and Interpolation
	5.1 Finite Differences
	5.2 Some More Examples of Finite Differences
	5.3 Error Propagation
	5.4 Numerical Unsatbility
	5.5 Interpolation
		(A) Newton’s Forward Difference Formula
		(B) Newton’s Backward Difference Formula
		(C) Central Difference Formulae
			(C1) Gauss’s Forward Interpolating Formula:
			(C2) Gauss’s Backward Interpolation Formula:
			(C3) Stirling’s Interpolation Formula:
			(C4) Bessel’s Interpolation Formula
			(C5) Everett’s Interpolation Formula
	5.6 Use of Interpolation Formulae
	5.7 Interpolation with Unequal-Spaced Points
		(A) Divided Differences
	5.8 Newton’s Fundamental (Divided Difference) Formula
	5.9 Error Formulae
	5.10 Lagrange’s Interpolation Formula
	5.11 Error in Lagrange’s Interpolation Formula
	5.12 Inverse Interpola Tion
		(A) Inverse Interpolation Using Newton’s Forward Difference Formula
		(B) Inverse Interpolation Using Everett’s Formula
		(C) Inverse Interpolation Using Lagrange’s Interpolation Formula
	Exercises
Chapter 6: Numerical Differentiation
	6.1 Centered Formula of Order O(h2 )
	6.2 Centered Formula of Order O(h2 )
	6.3 Error Analysis
		(A) Error for Centered Formula of Order O(h2 )
		(B) Error for Centered Formula of Order O(h4 )
	6.4 Richardson’s Extrapolation
	6.5 Central Difference Formula of Order O(h4 ) for fn(x)
	6.6 General Method for Deriving Differentiation Formulae
	6.7 Differentiation of a Function Tabulated in Unequal Intervals
	6.8 Differentiation of Lagrange’s Polynomial
	6.9 Differentiation of Newton Polynomial
	Exercises
Chapter 7: Numerical Quadrature
	7.1 General Quadrature Formula
		(A) Trapezoidal Rule:
		(B) Simpson’s One-Third Rule:
		(C) Simpson’s Three–Eight Rule:
		(D) Boole’s Rule:
		(E) Weddle’s Rule:
	7.2 Cote’s Formulae
	7.3 Error Term in Quadrature Formula
		Taylor’s Series Method for Finding Error
	7.4 Richardson Extrapolation (or Deferred Approach to the Limit)
	7.5 Simpson’s Formula with End Correction
	7.6 Romberg’s Method
	Exercises
Chapter 8: Ordinary Differential Equations
	8.1 Initial Value Problems and Boundary Value Problems
	8.2 Classification of Methods of Solution
	8.3 Single-Step Methods
		1. Taylor Series Method
		2. Euler’s Method
		3. Picard’s Method of Successive Integration
		4. Heun’s Method
		5. Runge–Kutta Method
		6. Runge–Kutta Method for System of First Order Equations
		7. Runge–Kutta Method for Higher Order Differential Equations
	Exercises
Appendix
	Model Paper I
	Model Paper II
	Statistical Tables
Index