Advanced Mathematics

Preface......Page 6 Acknowledgement......Page 8 Contents......Page 10 Unit l. Numerical Analysis-l......Page 16 1.2 Forward Differences......Page 18 1.3 Backward Differences......Page 19 1.4 Central Differences......Page 20 1.6 Relations between the Operators......Page 21 1.7 Fundamental theorem of the Difference Calculus......Page 22 1.9 To Show that......Page 23 1.10 To Show that......Page 24 Solved Examples......Page 25 Exercise 1.1......Page 33 Answers......Page 35 2.1 To Find One Missing Term......Page 36 2.2 Newton-Gregory's Formula for Forward Interpolation with Equal Intervals......Page 37 2.3 Newton-Gregory's Formula for Backward Interpolation with Equal Intervals......Page 38 2.4 Lagrange's Interpolation Formula for Unequal Intervals......Page 39 Solved Examples......Page 40 Exercise 2.1......Page 54 Answers......Page 56 3.1 Linear Equations......Page 57 3.2 Gauss Elimination Method......Page 58 3.3 Gauss-Seidel Method......Page 59 Solved Examples......Page 60 Answers......Page 66 4.3 Root of the Equation......Page 67 4.5 Regula-Falsi Method......Page 68 Solved Examples......Page 69 Answers......Page 77 5.2 Curve Fitting......Page 78 5.3 Method of Least Squares ......Page 79 Solved Examples......Page 80 Exercise 5.1......Page 85 Answers......Page 86 Unit ll Numerical Analysis-ll......Page 88 1.1 Derivatives Using Forward Difference Formula......Page 90 1.2 Derivatives Using Backward Difference Formula ......Page 91 1.3 Derivatives Using Stirling Difference Formula......Page 92 1.4 Derivatives Using Newton's Divided Difference Formula......Page 93 Solved Examples......Page 94 Exercise 1.1......Page 103 2.1 A General Quadrature Formula for Equally Spaced Arguments......Page 105 2.2 The Trapezoidal Rule......Page 106 2.4 Simpson's Three-Eighth Rule......Page 107 Solved Examples......Page 108 Exercise 2.1......Page 115 Answers......Page 116 3.1 Euler's Method......Page 117 3.3 Picard's Method of Successive Approximation......Page 118 3.4 Runge-Kutta Method ......Page 119 Solved Examples......Page 120 Exercise 3.1......Page 132 Answers......Page 133 4.2 Order of Difference Equation......Page 134 4.5 Fromation of Difference Equation......Page 135 4.6 Linear Difference Equation......Page 136 4.7 Homogeneous Linear Difference Equation with Constant Coefficient......Page 137 4.8 Non-Homogeneous Linear Difference Equation with Constant Coefficient......Page 138 Solved Examples......Page 139 Answers......Page 149 Unit lll. Special Function......Page 152 1.2 Solution of the Bessel's Functions......Page 154 1.3 General Solution of Bessel's Equation......Page 156 1.4 Integration of Bessel's Equations in Series for N=0......Page 157 1.5 Generating Function for Jn(X)......Page 158 1.6 Recurrence Relation for Jn(X)......Page 159 1.7 Orthogonal Property of Bessel's Functions......Page 161 Solved Examples......Page 163 Exercise 1.1......Page 169 2.2 Solution of Legendre's Equation......Page 171 2.4 Generating Function of Legendre's Polynomial Pn(X)......Page 174 2.5 Orthogonal Properties of Legendre's Polynomials......Page 175 2.6 Laplace's First Integral for Pn(X)......Page 177 2.7 Laplace's Second Integral for Pn(X)......Page 178 2.8 Rodrigue's Formula......Page 179 2.9 Recurrence Relations......Page 181 Solved Examples......Page 183 Exercise 2.1......Page 190 Answer ......Page 191 Unit lV. Statistics and Probability......Page 192 1.1 Terminology and Notations......Page 194 1.2 Definitions......Page 196 1.3 Elementary Theorems on Probability......Page 197 1.4 Addition Theorem of Probability......Page 199 Solved Examples......Page 200 Exercise 1.1......Page 202 1.5 Independent Events......Page 203 1.7 Multiplicative Theory of Probability or Theorem of Compound Probability......Page 204 Solved Examples......Page 205 Answers......Page 209 1.9 Baye's Theorem......Page 210 Solved Examples......Page 211 Answers......Page 216 1.12 Random Variable......Page 217 Solved Examples......Page 218 Exercise 1.4......Page 223 Answers......Page 224 2.1 Terminology and Notations ......Page 225 2.2 Binomial Distribution......Page 226 2.3 Constants of Binomial Distribution......Page 227 Solved Examples......Page 229 Exercise 2.1......Page 235 2.4 Poisson's Distribution......Page 237 2.5 Constants of Poisson Distribution ......Page 239 Solved Examples......Page 241 Exercise 2.2......Page 247 2.9 Constants of Normal Distribution......Page 249 2.10 Moment About the Mean M......Page 251 2.11 Area Under the Normal Curve......Page 252 2.12 Properties of the Normal Distribution and Normal Curve......Page 253 Solved Examples......Page 254 Answers......Page 262 3.2 Bivariate Frequency Distribution......Page 263 3.6 Linear or Non-Linear Correlation ......Page 264 3.9 Karl Pearson's Coefficients of Correlation ......Page 265 3.10 Probable Error......Page 269 Solved Examples......Page 270 3.12 Rank Correlation or Spearman's Coefficient of Rank Correlation......Page 275 3.14 Scatter Diagram or Dot Diagram......Page 276 Exercise 3.1......Page 282 3.15 Regression......Page 284 3.16 Line of Regression......Page 285 3.17 Standard Error of Estimate ......Page 286 Solved Examples......Page 287 Exercise 3.2......Page 293 Answers......Page 295 Unit V. Calculus of Variations and Transforms......Page 296 1.2 Euler's Equation......Page 298 1.3 Equivalent Forms of Euler's Equation......Page 300 1.4 Solution of Euler's Equations......Page 301 1.5 Strong and Weak Variations......Page 302 1.7 Variational Problems Involving Several Dependent Variables ......Page 303 1.8 Functionals Involving Second Order Derivatives......Page 304 Solved Examples......Page 305 Exercise 1.1......Page 316 Answers......Page 317 2.2 Linearity Properties......Page 318 2.4 Some Standard Z-Transforms......Page 319 2.5 Shifting Un to the Right......Page 321 2.7 Multiplication by n......Page 322 2.9 Initial Value Theorem......Page 323 Solved Examples......Page 324 Exercise 2.1......Page 330 Answers......Page 331 Solved Examples......Page 332 Answers......Page 335 2.13 Solution of Difference Equation by Z-Transform......Page 336 Answers......Page 338 About the Book: This book Àdvanced Mathematics ̀is primarily designed for B. Tech., IV Semester (EE and EC branch) students of Rajasthan Technical University. The subject matter is discussed in a lucid manner. The discussion is covered in five units: Unit I: deals with Numerical Analysis, Unit-II: gives different aspects of Numerical Analysis, Unit-III: Special Function, Unit-IV:Statistics and Probability, Calculus of Variation and Transforms are discussed in Unit V. All the theoretical concepts are explained through solved examples. Besides, a large number of unsolved problems on each top About the Book: This book 'Advanced Mathematics' is primarily designed for B. Tech., IV Semester (EE and EC branch) students of Rajasthan Technical University. The subject matter is discussed in a lucid manner. The discussion is covered in five units: Unit