Mathematical Handbook of Formulas and Tables,5E,M.R. Spiegel,2018

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Preface This handbook supplies a collection of mathematical formulas and tables which will be valuable to students and research workers in the fields of mathematics, physicS, engineering, and other sciences. Care has been taken to include only those formulas and tables which are most likely to be needed in practice, rather than highly spe cialized results which are rarely used. It is a"user-friendly" handbook with material mostly rooted in university mathematics and scientific courses. In fact. the first edition can already be found in many libraries and offices and it most likely has moved with the owners from office to office since their college times. Thus. this handbook has survived the test of time(while most other college texts have been thrown away) This new edition maintains the same spirit as previous editions, with the following changes. First of all we have deleted some out-of-date tables which can now be easily obtained from a simple calculator, and we have deleted sone rarely used formulas. The main change is that sections on Probability and Randon variables have been expanded with new material. These sections appear in both the physical and social sciences, including education There are also two new sections: Section Xlll on turing machines and section XIv on mathematica Finance Topics covered range from elementary to advanced Elementary topics include those from algebra, geom etry, trigonometry, analy tic geometry, probability and statistics, and calculus. Advanced topics include those from differential equations, numerical analysis, and vector analysis, such as Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, and elliptic and other special func tions of importance. This wide coverage of topics has been adopted to provide, within a single volume, most of the important mathematical results needed by student and research workers, regardless of their particular field of interest or level of attainment The book is divided into two main parts. Part a presents mathematical formulas together with other mate rial, such as definitions, theorems, graphs, diagrams, etc, essential for proper understanding and application of the formulas. Part B presents the numerical tables. These tables include basic statistical distributions(normal, Student's t, chi-square, etc. ) advanced functions (Bessel, Legendre, elliptic, etc. ) and financial functions(com pound and present value of an amount, and annuity) McGraw-Hill Education wishes to thank the various authors and publishers--for example, the Literary Executor of the late Sir Ronald A. Fisher, F.R.S., Dr. Frank Yates, F.R.S., and Oliver and Boyd Ltd, Edinburgh, for Table Ill of their book Statistical Tables for Biological, Agricultural and Medical research-who gave their permission to adapt data from their books for use in several tables in this handbook. Appropriate references to such sources are given below the corresponding tables Finally, I wish to thank the staff of McGraw-Hill Education Schaums Outline Series, especially Diane Grayson, for their unfailing cooperation SEYMOUR LIPSCHUTZ Temple University 00 Spiegel Math FM_ i-xii indc 5 109/1712:19pm Contents Part A FORMULAS 1 Section Elementary Constants, Products, Formulas 3 1. Greek Alphabet and Special Constants 2. Special Products and Factors 3. The binomial formula and Binomial coefficients 4. Complex Numbers 10 5. Solutions of Algebraic Equations 6. Conversion Factors 15 Section ll Geometry 16 7。 Geometric formulas 8. Formulas ron Plane Analytic Geometry 9. Special Plane Curves 10. Formulas from Solid Analytic Gcomctry 11. Special Moments of inertia Section Il Elementary Transcendental Functions 43 12. Trigonometric functions 43 13. Exponential and Logarithmic Functions 14. Hyperbolic functio 56 Section v Calculus 62 15. Derivatives 16. Indefinite Integrals 67 17. Tables of Special Indefinite Integrals 7 18. Definite Integrals 108 Section V Differential Equations and Vector Analysis 116 19. Basic Differential Equations and Solutions 116 20. Formulas froin Vector Analysis 119 Section vi Series 134 21. Series of Constants 22. Taylor Series 138 23. Bernoulli and euler numbers 142 24. Fourier series 00 Spiegel Math FM_i-xii indc 7 109/1712:19pm vIII CONTENTS Section VIl Special Functions and Polynomials 149 25. The gamma Function 149 26. The Bcta Function 27。 Bessel functions 30. Legendre and Associated Legendre Functions 28 9. Hermite polynomials 30. Lagucrrc and Associatcd Lagucrrc Polynomials 171 31. Chebyshev Polynomials 175 32. Hypergeometric Functions 178 Section Vlll Laplace and Fourier Transforms 180 33. Laplace Transforms 180 34. Fourier Trans forms 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 Section x Probability and statistics 208 39. Descriptive statistics 208 40. Probability 41. Random variables 223 Section xl Numerical methods 231 42. Interpolation 231 43. Quadrature 44. Solution of Nonlinear equations 237 45. Numerical Methods for Ordinary differential equations 239 40. Numerical Methods for Partial Differential Equations 24 47. Iteration Mcthods for Lincar Systcms 244 Section XIl Turing Machines 246 48. Basic Definitions, expressions 246 49。 Pictures 247 50. Quintuple, Turing machin 248 51. Computing with a Turing machine 52. Examples 52 Section xlv Mathematical finance 254 53. Basic probability 54。 Interest rates 256 55. Arbitrage Theorem and Options 00 Spiegel Math FM_i-xii indc 8 109/1712:19pm CONTENTS 56. Arbitrage Theorem 258 57. Black-Scholes formula 259 58. The Delta Hedging Arbitrage Strategy 260 Part B TABLES 263 Section I Logarithmic, Trigonometric, Exponential Functions 265 1. Four place common logarithms log nor log n 2. Sin x(x in Degrees and minutes) 267 3. Cos x(x in Degrees and minutes) 268 4. Tan x( in Dcgrccs and Minutes) 269 5. Conversion of Radians to Degrees, Minutes, and Seconds or Fractions of Degrees 6. Conversion of degrees Minutes and seconds to radians 271 7. Natural or Napierian Logarithms log x or In x 272 8. Exponential functions e 274 9. Exponential Functions e-r 275 10. Exponential, Sine, and Cosine integrals 276 Section l Factorial and Gamma Function Binomial coefficients 277 11. Factorial n 277 12.Gama上 unction 278 13. Binomial Coefficients 279 Section Bessel Functions 281 14. Bessel Functions . (x 15. Bessel Functions , (x) 16. Bcsscl Functions Y(r) 282 17. Bessel Functions Y,(x) 282 18. Bessel Functions I,(r) 283 19. Bessel Functions 1, (x) 20. Bessel Functions K(x) 28 21. Bessel Functions K(x 22. Bessel Functions Ber(x) 23. Bessel Functions Bei(r) 285 24. Bessel Functions Ker(x) 286 5. Bessel Functions Kei(x) 20. Values for Approximate Zeros of Bessel Functions 28 Section v Legendre Polynomials 288 27. Legendre Polynomials P(r) 288 28. Legendre Polynomials P (cos e) 289 Section V Elliptic Integrals 290 29. Conplete Elliptic Integrals of First and Second Kinds 290 30. Incomplete Elliptic Integral of the First Kind 291 31. Incomplctc Elliptic Intcgral of thc Sccond Kind 291 00 Spiegel Math FM_i-xii indc 9 109/1712:19pm X CONTENTS Section vi Financial Tables 292 32. Compound Amount: (I+r)" 33. Prcscnt Valuc of an Amount: (1 +r)- 293 34. Amount of an annuit (1+r) 294 35. Present value of an annuity (1+ Section Vl Probability and statistics 296 36. Areas Under the standard Normal curve fro -co lo x 296 37. Ordinates of the Standard Normal Curve 297 38. Percentile Values(t) for Students t Distribution 298 39. Percentile values(X Tor x(Chi-Square)Distribution 299 40. 95th Pcrccntilc Valucs for the F Distribution 300 41. 99th percentile values for the f distribution 301 42. Random numbers 302 Index of special Symbols and Notations 303 Index 305 00 Spiegel Math FM_i-xii indc 10 109/1712:19pm PART A FORMULAS 01 Spiegel section 1_001-015 indd 1 3003/1712:09pm Section / Elementary Constants, Products, Formulas GREEK ALPHABET and SPECIAL CONSTANTS Greek Alphabet Greek Greek letter Greek letter name name Lower case Capital Lower case Capital alpha A Nu Beta B XI Gamma r Omicron Delt ∏ E rho Zeta Sigma ∑ Eta 7 H Tau T Theta Upsilon Kappa K Lambda Mu Omega Special Constants 1.1.丌=3.141592653589793 L2.c=2.718281828459045…=lim1+ natural base of logarithms l3.y=0.577215664901532860606512.=Eler' s constant =lm1+-+-+…+--lnn 14.e=1.7810724179901979852..scl.3] 01 Spiegel section 1_001-015 indd 3 3003/1712:09pm GREEK ALPHABET AND SPECIAL CONSTANTS 15.√e=1.6487212707001281468 16.√z=r()=1.772453850905516027298167 where r is the gamma function [see 25.1 17.I(3)=2678938534707748 1.8.(x)=3625609908221908 1.9.1 radian=180°/=57.29577951308232. l.l0.1°=丌/180 radians=0.0174532925199432957692. radians 01 Spiegel section 1_001-015 indd 4 3003/1712:09pm

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