Mathematical Handbook of Formulas and Tables

Mathematical Handbook
of Formulas and Tables

More then 2400 Formulas and tables

Third Edition

Murray R. Spiegel, PhD
Former Professor and Chairman
Mathematics Department
Rensselaer Polytechnic Institute
Hartford Graduate Center

Seymour Lipschutz, PhD
Mathematics Department
Temple University

John Liu, PhD
Mathematics Department
University of Maryland

Schaum’s Outline Series

Contents

Section I Elementary Constants, Products, Formulas 3
1. Greek Alphabet and Special Constants 3
2. Special Products and Factors 5
3. The Binomial Formula and Binomial Coefficients 7
4. Complex Numbers 10
5. Solutions of Algebraic Equations 13
6. Conversion Factors 15

Section II Geometry 16
7. Geometric Formulas 16
8. Formulas from Plane Analytic Geometry 22
9. Special Plane Curves 28
10. Formulas from Solid Analytic Geometry 34
11. Special Moments of Inertia 41

Section III Elementary Transcendental Functions 43
12. Trigonometric Functions 43
13. Exponential and Logarithmic Functions 53
14. Hyperbolic Functions 56

Section IV Calculus 62
15. Derivatives 62
16. Indefinite Integrals 67
17. Tables of Special Indefinite Integrals 71
18. Definite Integrals 108

Section V Differential Equations and Vector Analysis 116
19. Basic Differential Equations and Solutions 116
20. Formulas from Vector Analysis 119

Section VI Series 134
21. Series of Constants 134
22. Taylor Series 138
23. Bernoulli and Euler Numbers 142
24. Fourier Series 144
Section VII Special Functions and Polynomials 149
25. The Gamma Function 149
26. The Beta Function 152
27. Bessel Functions 153
28. Legendre and Associated Legendre Functions 164
29. Hermite Polynomials 169
30. Laguerre and Associated Laguerre Polynomials 171
31. Chebyshev Polynomials 175
32. Hypergeometric Functions 178

Section VIII Laplace and Fourier Transforms 180
33. Laplace Transforms 180
34. Fourier Transforms 193

Section IX Elliptic and Miscellaneous Special Functions 198
35. Elliptic Functions 198
36. Miscellaneous and Riemann Zeta Functions 203

Section X Inequalities and Infinite Products 205
37. Inequalities 205
38. Infinite Products 207

Section XI Probability and Statistics 208
39. Descriptive Statistics 208
40. Probability 217
41. Random Variables 223

Section XII Numerical Methods 227
42. Interpolation 227
43. Quadrature 231
44. Solution of Nonlinear Equations 233
45. Numerical Methods for Ordinary Differential Equations 235
46. Numerical Methods for Partial Differential Equations 237
47. Iteration Methods for Linear Systems 240

Section I Logarithmic, Trigonometric, Exponential Functions 245
1. Four Place Common Logarithms log10 N or log N 245
2. Sin x (x in degrees and minutes) 247
3. Cos x (x in degrees and minutes) 248
4. Tan x (x in degrees and minutes) 249

5. Conversion of Radians to Degrees, Minutes, and Seconds
or Fractions of Degrees 250
6. Conversion of Degrees, Minutes, and Seconds to Radians 251
7. Natural or Napierian Logarithms loge x or ln x 252
8. Exponential Functions ex 254
9. Exponential Functions ex 255
10. Exponential, Sine, and Cosine Integrals 256

Section II Factorial and Gamma Function, Binomial Coefficients 257
11. Factorial n 257
12. Gamma Function 258
13. Binomial coefficients 259

Section III Bessel Functions 261
14. Bessel Functions J0(x) 261
15. Bessel Functions J1(x) 261
16. Bessel Functions Y0(x) 262
17. Bessel Functions Y1(x) 262
18. Bessel Functions I0(x) 263
19. Bessel Functions I1(x) 263
20. Bessel Functions K0(x) 264
21. Bessel Functions K1(x) 264
22. Bessel Functions Ber(x) 265
23. Bessel Functions Bei(x) 265
24. Bessel Functions Ker(x) 266
25. Bessel Functions Kei(x) 266
26. Values for Approximate Zeros of Bessel Functions 267

Section IV Legendre Polynomials 268
27. Legendre Polynomials Pn(x) 268
28. Legendre Polynomials Pn(cosx) 269

Section V Elliptic Integrals 270
29. Complete Elliptic Integrals of First and Second Kinds 270
30. Incomplete Elliptic Integral of the First Kind 271
31. Incomplete Elliptic Integral of the Second Kind 271

Section VI Financial Tables 272
32. Compound amount: (1 + r)^n 272
33. Present Value of an Amount: (1 + r)^(-n) 273
34. Amount of an Annuity: ((1+ r)^n –1)/r 274
35. Present Value of an Annuity: (1– (1+ r)^(-n))/r 275

Section VII Probability and Statistics 276
36. Areas Under the Standard Normal Curve 276
37. Ordinates of the Standard Normal curve 277
38. Percentile Values (tp) for Student's t Distribution 278
39. Percentile Values (?2p) for ?2 (Chi-Square) Distribution 279
40. 95th Percentile Values for the F distribution 280
41. 99th Percentile Values for the F distribution 281
42. Random Numbers 282

Index of Special Symbols and Notations 283
Index 285