MATHEMATICAL HANDBOOK. Formulas and Tables
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1 SCHAUM'S OUTLINE SERIES MATHEMATICAL HANDBOOK of Formulas and Tables Second Edition MURRAY R. SPIEGEL, Ph.D. Former Professor and Chairman Mathematics Department Rensselaer Polytechnic Institute Hartford Graduate Center JOHN LIU, Ph.D. Mathematics Department Temple University SCHAUM'S OUTLINE SERIES McGRAW-HILL New York San Francisco Washington, D.C. Auckland Bogota Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo Toronto
2 Contents Part A -,.,* $ ^^^Mmm^m^ FORMULAS Section 1: Elementary Constants, Products, Formulas 1. Greek Alphabet and Special Constants 1 2. Special Products and Factors 4 3. The Binomial Formula and Binomial Coefflcients 5 4. Complex Numbers 8 5. Solutions of Algebraic Equations Conversion Factors 12 Section II: Geometry 7. Geometrie Formulas Formulas from Plane Analytic Geometry Special Plane Curves Formulas from Solid Analytical Geometry Special Moments of Inertia 38 Section IM: Elementary Transcendental Functions 12. Trigonometrie Functions Exponential and Logarithmic Functions Hyperbolic Functions 53 Section IV: Calculus 15. Derivatives Indefinite Integrals Tables of Special Indefinite Integrals Definite Integrals 105 Section V: Differential Equations and Vector Analysis 19. Basic Differential Equations and Solutions Formulas from Vector Analysis 116
3 vi CONTENTS Section VI: Series 21. Series of Constants Taylor Series Bernoulli and Euler Numbers Fourier Series 141 Section VII: Special Functions and Polynomials 25. The Gamma Function The Beta Function Bessel Functions Legendre and Associated Legendre Functions Hermite Polynomials Laguerre and Associated Laguerre Polynomials Chebyshev Polynomials Hypergeometric Functions 176 Section VIII: Laplace and Fourier Transforms 33. Laplace Transforms Fourier Transforms 190 Section IX: Eliiptic and Miscellaneous Special Functions 35. Eliiptic Functions Miscellaneous and Riemann Zeta Functions 200 Section X: Inequalities and Infinite Products 37. Inequalities Infinite Products 204 Section XI: Probability and Statistics 39. Descriptive Statistics Random Variables Probability Distributions 216 Section XII: Numerical Methods 42. Interpolation Quadrature Solution of Nonlinear Equations Numerical Methods for Ordinary Differential Equations Numerical Methods for Partial Differential Equations Iteration Methods for Linear Systems 230
4 CONTENTS vii B Section I: Logarithmic, Trigonometrie, Exponential Functions 1. Four Place Common Logarithms Sina; (x in degrees and minutes) Cos«(x in degrees and minutes) Tanx (x in degrees and minutes) Conversion of Radians to Degrees, Minutes and Seconds Conversion of Degrees, Minutes and Seconds to Radians Natural or Napierian Logarithms log e a; or lnx Exponential Functions e* Exponential Functions e~* Exponential (Ei), Sine (Si) and Cosine (Ci) Integrals 243 Section II: Factorial and Gamma Function, Binomial Coefficients 11. Factorial n Gamma Function Binomial Coefficients 246 Section III: Bessel Functions 14. Bessel Functions J 0 {x) Bessel Functions J x (x) Bessel Functions Y 0 (x) Bessel Functions Y^x) Bessel Functions I 0 (x) Bessel Functions I r (x) Bessel Functions K 0 {x) Bessel Functions K±(x), Bessel Functions Ber(«) Bessel Functions Bei(x) Bessel Functions Ker(x) Bessel Functions Kei(x) Values for Approximate Zeros of Bessel Functions 254 Section IV: Legendre Polynomials 27. Legendre Polynomials P n {x) Legendre Polynomials P (cos0) 256
5 viü CONTENTS Section V: Elliptic Integrals 29. Complete Elliptic Integrals of First and Second Kinds Incomplete Elliptic Integrals of the First Kind Incomplete Elliptic Integrals of the Second Kind 258 Section VI: Financial Tables 32. Compound Amount: (1 + r) n Present Value of an Amount: (1 + r)~" 260 (1 + r) M Amount of an Annuity: (1 + r)~" 35. Present Value of an Annuity: 262 Section VII: Probability and Statistics 36. Areas under the Standard Normal Curve Ordinates of the Standard Normal Curve Percentile Values for Student's t Distribution Percentile Values for x 2 (Chi-Square) Distribution th Percentile Values for the F Distribution th Percentile Values for the F Distribution Random Numbers 269 Index of Special Symbols and Notations 271 Index 273
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