Schaums mathematical handbook of formulas and tables

P r e f The pur-pose of this handbook is to supply a collection of mathematical formulas and tables which will prove to be valuable to students and research workers in the fields of mathematics, physics, engineering and other sciences TO accomplish this, tare has been taken to include those formulas and tables which are most likely to be needed in practice rather than highly specialized results which are rarely used Every effort has been made to present results concisely as well as precisely SOthat they may be referred to with a maximum of ease as well as confidence Topics covered range from elementary to advanced Elementary topics include those from algebra, geometry, trigonometry, analytic geometry and calculus Advanced topics include those from differential equations, vector analysis, Fourier series, gamma and beta functions, Bessel and Legendre functions, Fourier and Laplace transforms, elliptic functions and various other special functions of importance This wide coverage of topics has been adopted SOas to provide within a single volume most of the important mathematical results needed by the student or research worker regardless of his particular field of interest or level of attainment The book is divided into two main parts Part presents mathematical formulas together with other material, such as definitions, theorems, graphs, diagrams, etc., essential for proper understanding and application of the formulas Included in this first part are extensive tables of integrals and Laplace transforms which should be extremely useful to the student and research worker Part II presents numerical tables such as the values of elementary functions (trigonometric, logarithmic, exponential, hyperbolic, etc.) as well as advanced functions (Bessel, Legendre, elliptic, etc.) In order to eliminate confusion, especially to the beginner in mathematics, the numerical tables for each function are separated, Thus, for example, the sine and cosine functions for angles in degrees and minutes are given in separate tables rather than in one table SOthat there is no need to be concerned about the possibility of errer due to looking in the wrong column or row wish to thank the various authors and publishers who gave me permission to adapt data from their books for use in several tables of this handbook Appropriate references to such sources are given next to the corresponding tables In particular am indebted to the Literary Executor of the late Sir Ronald A Fisher, F.R.S., to Dr Frank Yates, F.R.S., and to Oliver and Boyd Ltd., Edinburgh, for permission to use data from Table III of their book S T tf B a Aao i b a gtMy o R l n ir e l e e d si d also wish to express my gratitude to Nicola Menti, Henry Hayden and Jack Margolin for their excellent editorial cooperation M R SPIEGEL Rensselaer Polytechnic Institute September, 1968 o s s tc i CONTENTS Page Special Constants Special Products and Factors The Binomial Formula and Binomial Coefficients Geometric Formulas 5 Trigonometric Functions 11 Complex Numbers 21 Exponential and Logarithmic Functions 23 Hyperbolic Functions 26 Solutions of Algebraic Equations 32 10 Formulas from Plane Analytic Geometry 34 40 11 Special Plane Curves ~ 12 Formulas from Solid Analytic Geometry 46 13 Derivatives 53 14 Indefinite Integrals 57 15 Definite Integrals 94 16 The Gamma Function 10 17 The Beta Function 18 Basic Differential Equations and Solutions 19 Series of Constants lO 20 Taylor Series ll 21 Bernoulliand 22 Formulas from Vector Analysis 23 Fourier Series ~3 24 Bessel Functions 13 2s Legendre Functions l4 26 Associated Legendre Functions 149 27 28 Hermite Polynomials l5 Laguerre Polynomials 153 29 Associated Laguerre Polynomials 30 Chebyshev Polynomials l5 Euler Numbers lO 104 114 116 KG Part I FORMULAS THE GREEK Greek name G&W ALPHABET Greek name Greek Lower case tter Capital Alpha A Nu N Beta B Xi sz Gamma l? Omicron Delta A Pi IT Epsilon E Rho P Zeta Z Sigma Eta H Tau T Theta (3 Upsilon k Iota Phi @ Kappa K Chi X Lambda A Psi * MU M Omega n 1.1 1.2 = natural base of logarithms 1.3 fi = 1.41421 35623 73095 04889 1.4 fi = 1.73205 08075 68877 2935 1.5 fi = 2.23606 79774 1.6 h = 1.25992 1050 1.7 & = 1.44224 9570 1.8 fi = 1.14869 8355 1.9 b = 1.24573 0940 1.10 eT = 23.14069 26327 79269 006 1.11 re = 22.45915 77183 61045 47342 715 1.12 ee = 22414 1.13 logI,, = 0.30102 99956 63981 19521 37389 1.14 logI,, = 0.47712 12547 19662 43729 50279 1.15 logIO e = 0.43429 44819 03251 82765 1.16 logul ?r = 0.49714 98726 94133 85435 12683 1.17 loge 10 In 10 1.18 loge = ln = 0.69314 71805 59945 30941 1.19 loge = ln = 1.09861 22886 68109 1.20 y = 1.21 ey = 1.22 fi = 1.23 = 15.15426 = 0.57721 56649 1.78107 r(&) = 79264 2.30258 190 12707 6512 9852 00128 1468 1.77245 2.67893 85347 07748 1.25 r(i) 3.62560 99082 21908 1-26 radian 1.27 1” = ~/180 radians = = 57.29577 0.01745 7232 69139 5245 = Eukr's co%stu~t [see 1.201 38509 05516 II’(&) = 180°/7r 02729 ~ZLYLC~~OTZ [sec pages 1.24 = 50929 94045 68401 7991 01532 86060 F is the gummu = 99789 6964 24179 90197 1.64872 where = 8167 .O 95130 8232 32925 101-102) 19943 29576 92 radians Index of Special Symbols and Notations The following list shows special symbols and notations used in this book together with pages on which Cases where a symbol has more than one meaning will be clear from they are defined or first appear the context Symbole Berri (x), Bein (xj B(m, n) 4l (34 Ci(x) e elp e2, e3 natural Euler 7, T-l Fourier elliptic Hermite in curvilinear unit vectors In(x) modified Jr, (4 Bessel in rectangular Bessel function function complete kind, 138 coordinates, 117 integral 136 of first kind, 179 140 Bessel or loge x natural logarithm common function polynomials, and inverse pn (4 Legendre f%4 associated Qn (4 Legendre Qt’b) associated Legendre cylindrical coordinate, polynomials, functions kind, 148 functions of second 22, 36 sine integral, 50 184 183 polynomials of first kind, 157 Chebyshev polynomials of second function kind, 49 Chebyshev Bessel transform, of first kind, 149 of second coordinate, sine integral, 155 Laplace 146 Legendre functions coordinate, Fresnel 153 Laguerre transform spherical kind, 139 of x, 23 polynomials, Laplace polar of second of x, 24 logarithm Laguerre associated r 175, 176 124 of first kind, 138 of first kind, elliptic modified L?(x) transform, 151 of first and second Wr) B A is greater than B [or B is less than A] A