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