www.TechnicalBooksPDF.com HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition www.TechnicalBooksPDF.com This Page Intentionally Left Blank www.TechnicalBooksPDF.com HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom San Diego San Francisco New York Boston London Sydney Tokyo www.TechnicalBooksPDF.com This book is printed on acid-free paper ∞ Copyright C 2000, 1995 by Academic Press All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher Requests for permission to make copies of any part of the work should be mailed to the following address: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777 ACADEMIC PRESS 525 B Street, Suite 1900, San Diego, CA 92101-4495 USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY UK Library of Congress Catalog Number: 95-2344 International Standard Book Number: 0-12-382251-3 Printed in the United States of America 00 01 02 03 04 COB www.TechnicalBooksPDF.com Contents Preface xix Preface to the Second Edition xxi Index of Special Functions and Notations xxiii Quick Reference List of Frequently Used Data 0.1 Useful Identities 0.1.1 Trigonometric identities 0.1.2 Hyperbolic identities 0.2 Complex Relationships 0.3 Constants 0.4 Derivatives of Elementary Functions 0.5 Rules of Differentiation and Integration 0.6 Standard Integrals 0.7 Standard Series 11 0.8 Geometry 13 Numerical, Algebraic, and Analytical Results for Series and Calculus 1.1 Algebraic Results Involving Real and Complex Numbers 25 1.1.1 Complex numbers 25 1.1.2 Algebraic inequalities involving real and complex numbers 26 v www.TechnicalBooksPDF.com vi Contents 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 Finite Sums 29 1.2.1 The binomial theorem for positive integral exponents 29 1.2.2 Arithmetic, geometric, and arithmetic–geometric series 33 1.2.3 Sums of powers of integers 34 1.2.4 Proof by mathematical induction 36 Bernoulli and Euler Numbers and Polynomials 37 1.3.1 Bernoulli and Euler numbers 37 1.3.2 Bernoulli and Euler polynomials 43 1.3.3 The Euler–Maclaurin summation formula 45 1.3.4 Accelerating the convergence of alternating series 46 Determinants 47 1.4.1 Expansion of second- and third-order determinants 47 1.4.2 Minors, cofactors, and the Laplace expansion 48 1.4.3 Basic properties of determinants 50 1.4.4 Jacobi’s theorem 50 1.4.5 Hadamard’s theorem 51 1.4.6 Hadamard’s inequality 51 1.4.7 Cramer’s rule 52 1.4.8 Some special determinants 52 1.4.9 Routh–Hurwitz theorem 54 Matrices 55 1.5.1 Special matrices 55 1.5.2 Quadratic forms 58 1.5.3 Differentiation and integration of matrices 60 1.5.4 The matrix exponential 61 1.5.5 The Gerschgorin circle theorem 61 Permutations and Combinations 62 1.6.1 Permutations 62 1.6.2 Combinations 62 Partial Fraction Decomposition 63 1.7.1 Rational functions 63 1.7.2 Method of undetermined coefficients 63 Convergence of Series 66 1.8.1 Types of convergence of numerical series 66 1.8.2 Convergence tests 66 1.8.3 Examples of infinite numerical series 68 Infinite Products 71 1.9.1 Convergence of infinite products 71 1.9.2 Examples of infinite products 71 Functional Series 73 1.10.1 Uniform convergence 73 Power Series 74 1.11.1 Definition 74 www.TechnicalBooksPDF.com vii Contents 1.12 Taylor Series 79 1.12.1 Definition and forms of remainder term 79 1.12.4 Order notation (Big O and little o) 80 1.13 Fourier Series 81 1.13.1 Definitions 81 1.14 Asymptotic Expansions 85 1.14.1 Introduction 85 1.14.2 Definition and properties of asymptotic series 86 1.15 Basic Results from the Calculus 86 1.15.1 Rules for differentiation 86 1.15.2 Integration 88 1.15.3 Reduction formulas 91 1.15.4 Improper integrals 92 1.15.5 Integration of rational functions 94 1.15.6 Elementary applications of definite integrals 96 Functions and Identities 2.1 Complex Numbers and Trigonometric and Hyperbolic Functions 101 2.1.1 Basic results 101 2.2 Logarithms and Exponentials 112 2.2.1 Basic functional relationships 112 2.2.2 The number e 113 2.3 The Exponential Function 114 2.3.1 Series representations 114 2.4 Trigonometric Identities 115 2.4.1 Trigonometric functions 115 2.5 Hyperbolic Identities 121 2.5.1 Hyperbolic functions 121 2.6 The Logarithm 126 2.6.1 Series representations 126 2.7 Inverse Trigonometric and Hyperbolic Functions 128 2.7.1 Domains of definition and principal values 128 2.7.2 Functional relations 128 2.8 Series Representations of Trigonometric and Hyperbolic Functions 133 2.8.1 Trigonometric functions 133 2.8.2 Hyperbolic functions 134 2.8.3 Inverse trigonometric functions 134 2.8.4 Inverse hyperbolic functions 135 2.9 Useful Limiting Values and Inequalities Involving Elementary Functions 136 2.9.1 Logarithmic functions 136 2.9.2 Exponential functions 136 2.9.3 Trigonometric and hyperbolic functions 137 www.TechnicalBooksPDF.com viii Contents Derivatives of Elementary Functions 3.1 3.2 3.3 3.4 3.5 Derivatives of Algebraic, Logarithmic, and Exponential Functions Derivatives of Trigonometric Functions 140 Derivatives of Inverse Trigonometric Functions 140 Derivatives of Hyperbolic Functions 141 Derivatives of Inverse Hyperbolic Functions 142 139 Indefinite Integrals of Algebraic Functions 4.1 Algebraic and Transcendental Functions 145 4.1.1 Definitions 145 4.2 Indefinite Integrals of Rational Functions 146 4.2.1 Integrands involving x n 146 4.2.2 Integrands involving a + bx 146 4.2.3 Integrands involving linear factors 149 4.2.4 Integrands involving a ± b2 x 150 4.2.5 Integrands involving a + bx + cx 153 4.2.6 Integrands involving a + bx 155 4.2.7 Integrands involving a + bx 156 4.3 Nonrational Algebraic Functions 158 √ 4.3.1 Integrands containing a + bx k and x 158 4.3.2 Integrands containing (a + bx)1/2 160 4.3.3 Integrands containing (a + cx )1/2 161 4.3.4 Integrands containing (a + bx + cx )1/2 164 Indefinite Integrals of Exponential Functions 5.1 Basic Results 167 5.1.1 Indefinite integrals involving eax 167 5.1.2 Integrands involving the exponential functions combined with rational functions of x 168 5.1.3 Integrands involving the exponential functions combined with trigonometric functions 169 Indefinite Integrals of Logarithmic Functions 6.1 Combinations of Logarithms and Polynomials 173 6.1.1 The logarithm 173 6.1.2 Integrands involving combinations of ln(ax) and powers of x 6.1.3 Integrands involving (a + bx)m lnn x 175 www.TechnicalBooksPDF.com 174 ix Contents 6.1.4 Integrands involving ln(x ± a ) 177 6.1.5 Integrands involving x m ln[x + (x ± a )1/2 ] Indefinite Integrals of Hyperbolic Functions 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 178 Basic Results 179 7.1.1 Integrands involving sinh(a + bx) and cosh(a + bx) 179 Integrands Involving Powers of sinh(bx) or cosh(bx) 180 7.2.1 Integrands involving powers of sinh(bx) 180 7.2.2 Integrands involving powers of cosh(bx) 180 Integrands Involving (a ± bx)m sinh(cx) or (a + bx)m cosh(cx) 181 7.3.1 General results 181 Integrands Involving x m sinhn x or x m coshn x 183 7.4.1 Integrands involving x m sinhn x 183 7.4.2 Integrals involving x m coshn x 183 Integrands Involving x m sinh−n x or x m cosh−n x 183 7.5.1 Integrands involving x m sinh−n x 183 7.5.2 Integrands involving x m cosh−n x 184 Integrands Involving (1 ± cosh x)−m 185 7.6.1 Integrands involving (1 ± cosh x)−1 185 7.6.2 Integrands involving (1 ± cosh x)−2 185 Integrands Involving sinh(ax)cosh−n x or cosh(ax)sinh−n x 185 7.7.1 Integrands involving sinh(ax) cosh−n x 185 7.7.2 Integrands involving cosh(ax) sinh−n x 186 Integrands Involving sinh(ax + b) and cosh(cx + d) 186 7.8.1 General case 186 7.8.2 Special case a = c 187 7.8.3 Integrands involving sinh p xcoshq x 187 Integrands Involving kx and coth kx 188 7.9.1 Integrands involving kx 188 7.9.2 Integrands involving coth kx 188 Integrands Involving (a + bx)m sinh kx or (a + bx)m cosh kx 189 7.10.1 Integrands involving (a + bx)m sinh kx 189 7.10.2 Integrands involving (a + bx)m cosh kx 189 Indefinite Integrals Involving Inverse Hyperbolic Functions 8.1 Basic Results 191 8.1.1 Integrands involving products of x n and arcsinh(x/a) or arccosh(x/a) 191 8.2 Integrands Involving x −n arcsinh(x/a) or x −n arccosh(x/a) 193 8.2.1 Integrands involving x −n arcsinh(x/a) 193 8.2.2 Integrands involving x −n arccosh(x/a) 193 www.TechnicalBooksPDF.com 420 Short Classified Reference List Erd´elyi, A., Asymptotic Expansions, Dover Publications, New York, 1956 Olver, F W J., Asymptotics and Special Functions, Academic Press, New York, 1974 Elliptic integrals Abramowitz, M., and Stegun, I A., Handbook of Mathematical Functions, Dover Publications, New York, 1972 Byrd, P F., and Friedman, M D., Handbook of Elliptic Integrals for Engineers and Physicists, SpringerVerlag, Berlin, 1954 Gradshteyn, I S., and Ryzhik, I M., Tables of Integrals, Series, and Products, (A Jeffrey, Ed.), 5th ed., Academic Press, Boston, 1994 Lawden, D F., Elliptic Functions and Applications, Springer-Verlag, Berlin, 1989 Neville, E H., Jacobian Elliptic Functions, 2nd ed., Oxford University Press, Oxford, 1951 Prudnikov., A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 3, Gordon and Breach, New York, 1990 Integral transforms Doetsch, G., Handbuch der Laplace-Transformation, Vols IIV, Birkhăauser Verlag, Basel, 195056 Doetsch, G., Theory and Application of the Laplace Transform, Chelsea, New York, 1965 Erd´elyi, A., et al., Tables of Integral Transforms, Vols I and II, McGraw–Hill, New York, 1954 Jury, E I., Theory and Application of the z-Transform Method, Wiley, New York, 1964 Marichev, O I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1982 Oberhettinger, F., and Badii, L., Tables of Laplace Transforms, Springer-Verlag, Berlin, 1973 Oppenheim, A V., and Schafer, R W., Discrete Signal Processing, Prentice Hall, New York, 1989 Prudnikov, A P., Brychkov, Yu A., and Marichev, O I., Integrals and Series, Vol 4, Gordon and Breach, New York, 1992 Sneddon, I N., Fourier Transforms, McGraw–Hill, New York, 1951 Sneddon, I N., The Use of Integral Transforms, McGraw–Hill, New York, 1972 Widder, D V., The Laplace Transforms, Princeton University Press, Princeton, NJ, 1941 Orthogonal functions and polynomials Abramowitz, M., and Stegun, I A., Handbook of Mathematical Functions, Dover Publications, New York, 1972 Sansone, G., Orthogonal Functions, revised English ed., Interscience, New York, 1959 Szegăo, G., Orthogonal Polynomials, revised ed., Colloquium Publications XXIII, American Mathematical Society, New York, 1959 Series Jolley, I R W., Summation of Series, Dover Publications, New York, 1962 Zygmund, A., Trigonometric Series, 2nd ed., Vols I and II, Cambridge University Press, London, 1988 Numerical tabulations and approximations Abramowitz, M., and Stegun, I A., Handbook of Mathematical Functions, Dover Publications, New York, 1972 Hastings, Jr., C., Approximations for Digital Computers, Princeton University Press, Princeton, NJ, 1955 Jahnke, E., and Emde, F., Tables of Functions with Formulas and Curves, Dover Publications, New York, 1943 Jahnke, E., Emde, F., and Lăosch, F., Tables of Higher Functions, 6th ed., McGrawHill, New York, 1960 Ordinary and partial differential equations Birkhoff, G., and Gian–Carlo, R., Ordinary Differential Equations, 4th ed., Wiley, New York, 1989 www.TechnicalBooksPDF.com 421 Short Classified Reference List Boyce, W E., and Di Prima, R C., Elementary Differential Equations and Boundary Value Problems, 5th ed., Wiley, New York, 1992 Du Chateau, Y., and Zachmann, D., Applied Partial Differential Equations, Harper & Row, New York, 1909 Keener, J P., Principles of Applied Mathematics, Addison–Wesley, New York, 1988 Logan, J D., Applied Mathematics: A Contemporary Approach, Wiley, New York, 1987 Strauss, W A., Partial Differential Equations, Wiley, New York, 1992 Tyn, Myint-U., Partial Differential Equations of Mathematical Physics, Elsevier, New York, 1973 Zachmanoglou, E C., and Thoe, D W., Introduction to Partial Differential Equations and Applications, William and Wilkins, 1976 Zauderer, E., Partial Differential Equations of Applied Mathematics, 2nd ed., Wiley, New York, 1989 Zwillinger, D., Handbook of Differential Equations, Academic Press, New York, 1989 Numerical analysis Ames, W F., Nonlinear Partial Differential Equations in Engineering, Vol 1, Academic Press, New York, 1965 Ames, W F., Nonlinear Partial Differential Equations in Engineering, Vol 2, Academic Press, New York, 1972 Ames, W F., Numerical Methods for Partial Differential Equations, Nelson, London, 1977 Atkinson, K E., An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989 Frăoberg, C E., Numerical Methods: Theory and Computer Applications, Addison–Wesley, New York, 1985 Golub, G H., and Van Loan, C E., Matrix Computations, Johns Hopkins University Press, Baltimore, 1984 Henrici, P., Essentials of Numerical Analysis, Wiley, New York, 1982 Johnson, L W., and Riess, R D., Numerical Analysis, Addison–Wesley, New York, 1982 Morton, K W., and Mayers, D F., Numerical Solution of Partial Differential Equations, Cambridge University Press, London, 1994 Press, W H., Flannery, B P., Teukolsky, S A., and Vellerling, W T., Numerical Recipes, Cambridge University Press, London, 1986 Schwarz, H R., Numerical Analysis: A Comprehensive Introduction, Wiley, New York, 1989 www.TechnicalBooksPDF.com This Page Intentionally Left Blank www.TechnicalBooksPDF.com Index A Abel’s test, for convergence, 68, 94 Abel’s theorem, 74 Absolute convergence infinite products, 71 series, 66 Absolute value complex number, 26 integral quality, 90 Acceleration of convergence, 46–47 Addition, see Sum Adjoint matrix, 56 Advection equation, 393 Algebraic function definition, 145 derivatives, 139–140 indefinite integrals, 5–7, 146–165 nonrational, see Nonrational algebraic functions rational, see Rational algebraic functions Algebraic jump condition, 397 Algebraically homogeneous differential equation, 326 Alternant, 52 Alternating series, 45, 68 Amplitude, of Jacobian elliptic function, 235 Annulus geometry, 10 Annulus, of circle, 18–19 Antiderviative, 88, 364, see also Indefinite integral Arc circular, 17 length, 98 line integral along, 364 Area under a curve, 96 geometric figures, 13–24 surface, 366 surface of revolution, 98–99 Argument, of complex number, 101 Arithmetic–geometric inequality, 28 Arithmetic–geometric series, 33, 68 Arithmetic series, 33 Associative laws, of vector operations, 356, 358, 359 Asymptotic expansions definition, 81–82 error function, 243 normal distribution, 241–242 order notation, 77 Asymptotic representation Bernoulli numbers, 42–43 Bessel functions, 273, 278 gamma function, 223 n!, 223 Asymptotic series, 85–86 B Bernoulli number asymptotic relationships for, 42–43 definitions, 37, 39 list of, 38, 39 occurrence in series, 40–41 relationships with Euler numbers and polynomials, 39–40 series representations for, 40 sums of powers of integers, 34 Bernoulli polynomials, 37, 40, 43–44 Bernoulli’s equation, 324 Bessel functions, 269–284 asymptotic representations, 273, 278 definite integrals, 282–283 of fractional order, 272–273, 277–278 graphs, 274, 275 indefinite integrals, 281–282 423 www.TechnicalBooksPDF.com 424 Index Bessel functions (continued ) integral representations, 281 modified, 274–278 of first and second kind, 275 relationships between, 278–280 series expansions, 270–272, 276–277 spherical, 283, 284 zeros, 273 Bessel’s equation forms of, 269–270, 337–339 partial differential equations, 389 Bessel’s inequality, 84 Bessel’s modified equation, 274–275, 337–338 Beta function, 224–225 Big O, 80 Binomial coefficients, 29 definition, 30 generation, 30–31 permutations, 62 relationships between, 31 sums of powers of integers, 31–33 table, 30 Binomial expansion, 11, 68–69 Binomial series, 11, 68–69 Binomial theorem, 29–33 Bipolar coordinates, 374–375 Bisection method, for determining roots, 105–106 Bode’s rule, for numerical integration, 317 Boundary conditions, 350 Boundary value problem ordinary differential equations, 327, 350–352 partial differential equations, 381–382, 383 Bounded variation, 82 Burger’s equation, 400 Burger’s shock wave, 401–402 C Cardano formula, 100 Carleman’s inequality, 28–29 Cartesian coordinates, 354–355, 372 Cartesian representation, of complex number, 102 Cauchy condition, for partial differential equations, 383 Cauchy criterion, for convergence, 66, 73 Cauchy–Euler equation, 336–337 Cauchy form, of Taylor series remainder term, 79 Cauchy integral test, 67 Cauchy nth root test, 67 Cauchy principal value, 93 Cauchy problem, 383 Cauchy–Schwarz–Buniakowsky inequality, 27 Cayley–Hamilton theorem, 58 Center of mass (gravity), 99 Centroid, 13–23, 99 Chain rule, 87, 363 Characteristic curve, 392 Characteristic equation, 57, 58 Characteristic form, of partial differential equation, 392–393 Characteristic method, for partial differential equations, 392–396 Characteristic parameter, 230 Characteristic polynomial, 327 Chebyshev polynomial, 290–294, 410–412 Chebyshev’s inequality, 28 Circle geometry, 17–18 Circle of convergence, 75 Circulant, 52–53 Circumference, 17 Closed-form summation, 266–268 Closed-type integration formula, 315–320 Coefficients binomial, see Binomial coefficient Fourier, 81, 257–265 multinomial, 62 polynomial, 104–105 undetermined, 63–64 Cofactor, 48 Combinations, 62–63 Commutative laws, of vector operations, 356, 358 Comparison integral inequality, 90 Comparison test, for convergence, 67 Compatibility condition, for partial differential equations, 384 Complementary error function, 242 Complementary function, linear differential equation, 326 Complementary minor, 50–51 Complementary modulus, 230 Complete elliptic integral, 230 Complex conjugate, 26 Complex Fourier series, 83, 260–261, 265 Complex numbers Cartesian representation, 102 conjugate, 26 www.TechnicalBooksPDF.com de Moivre’s theorem, 102 definitions, 25–26, 101–102 difference, 25 equality, 25 Euler’s formula, 102 identities, 2, 102–103 imaginary part, 26 imaginary unit, 25 inequalities, 27, 28 modulus, 26 modulus-argument form, 101 principal value, 102 quotient, 26 real part, 26 roots, 103–104 sums of powers of integers, 25 triangle inequality, 26 Components, of vectors, 354, 357–358 Composite integration formula, 316 Compressible gas flow, 391 Computational molecule, 416 Conditionally convergent series, 66 Cone geometry, 20–22 Confluent hypergeometric equation, 345 Conservation equation, 391–392 Conservative field, 365 Constant e, Euler–Mascheroni, 222 Euler’s, 222 gamma function as, of integration, 88 log10 e, method of undetermined coefficients, 334 pi, Constitutive equation, 391 Contact discontinuity, 397 Convergence acceleration of, 45 of functions, see Roots of functions improper integrals, 92–94 infinite products, 71 Convergence of series absolute, 66 Cauchy criterion for, 66 divergence, 66 Fourier, 81–82, 257–268 infinite products, 71–72 partial sum, 66 power, 74–75 Taylor, 79–80 tests, 66–68 425 Index Abel’s test, 68 alternating series test, 45, 67 Cauchy integral test, 67 Cauchy nth root test, 67 comparison test, 67 Dirichlet test, 68 limit comparison test, 67 Raabe’s test, 67 types of, 63 uniform convergence, 73–74 Convex function, 28–29 Convexity, 29 Coordinates bipolar, 374–375 Cartesian, 354–355, 372 curvilinear, 369–371 cylindrical, 372, 376, 377–378 definitions, 369–371 elliptic cylinder, 377–378 oblate spheroidal, 379 orthogonal, 369–379 parabolic cylinder, 376 paraboloidal, 377 polar, 372 prolate spheroidal, 378–379 rectangular, 372 spherical, 372–374 spheroidal, 378–379 toroidal, 378–379 Cosine Fourier series, 259, 264 Cosine Fourier transform, 309, 311–314 Cosine integrals, 245–248 Cramer’s rule, 52 Cross product, 359–360 Cube, 14 Cubic equation, 100 Cubic spline interpolation, 410 Curl, 367, 371 Curvilinear coordinates, see Coordinates Cylinder geometry, 19–20 Cylindrical coordinates, 372, 376, 377–378 Cylindrical wave equation, 400 D D’Alembert’s ratio test, 66 De Moivre’s theorem, 102–103 Definite integral applications, 96–99 Bessel functions, 282–283 definition, 88 exponential function, 254–255 hyperbolic functions, 256 involving powers of x, 249–251 logarithmic function, 256 trigonometric functions, 250–254 vector functions, 364 Delta amplitude, 235 Delta function, 142, 404 Derivative algebraic functions, 139–140 approximation to, 414 directional, 368 error function, 243 exponential function, 3, 140 Fourier series, 84 function of a function, hyperbolic functions, 141–142 inverse hyperbolic functions, 142–143 inverse trigonometric functions, 140–141 Jacobian elliptic function, 237 Laplace transform of, 300 logarithmic function, 3, 139–140 matrix, 60–61 power series, 75 trigonometric functions, 3, 140 vector functions, 361–363 Determinant alternant, 52 basic properties, 50 circulant, 52–53 cofactors, 48 Cramer’s rule, 52 definition, 47–48 expansion of, 47–50 Hadamard’s inequality, 51 Hadamard’s theorem, 51 Hessian, 53–54 Jacobian, 53, 369–370 Jacobi’s theorem, 50–51 Laplace expansion, 48 minor, 48 Routh–Hurwitz theorem, 54–55 Vandermonde’s, 52 Wronskian, 5, 322–323 Diagonal matrix, 55, 57 Diagonally dominant matrix, 51, 57 Diagonals, of geometric figures, 13–15 Difference equations z-transform and, 408 numerical methods for, 409–418 Differential equations Bessel’s, see Bessel functions; Bessel’s equation Chebyshev polynomials, 290–291 Hermite polynomials, 296 Laguerre polynomials, 294 www.TechnicalBooksPDF.com Legendre polynomials, 286, 289, 337 ordinary, see Ordinary differential equations partial, see Partial differential equations solution methods, 327–352, 385–402 Differentiation elementary functions, 3, 139 exponential function, 139 hyperbolic functions, 141 integral containing a parameter, inverse hyperbolic functions, 142 inverse trigonometric functions, 140 logarithmic functions, 139 product, 3, 87 quotient, 3, 87 rules of, 3, 86–87 sums of powers of integers, 3, 87 term by term, 74 trigonometric functions, 140 Digamma function, 224 Dini’s condition, for Fourier series, 82 Dirac delta function, 301 Direction cosine, 354 Direction ratio, 355 Directional derivative, 368 Dirichlet condition Fourier series, 82 Fourier transform, 307 partial differential equations, 383, 394 Dirichlet integral representation, 83–84 Dirichlet kernel, 84 Dirichlet problem, 384 Dirichlet’s result, 82 Dirichlet’s test, for convergence, 68, 94 Dirichlet’s theorem, 74 Discontinuous functions, and Fourier series, 265–268 Discontinuous solution, to partial differential equation, 396–398 Discriminant, 329 Dispersive effect, 401 Dissipative effect, 400 Distributive laws, of vector operations, 356, 358, 359 Divergence infinite products, 71 vectors, 366–367, 371 Divergence form, of conservation equation, 392 Divergent series, 66–67, 85 Division algorithm, 105 Dot product, 358–359 426 Index Double arguments, in Jacobian elliptic function, 236 Dummy variable, for integration, 88 E e, see also Exponential function constant, definitions, 113 numerical value, 2, 113 series expansion for, 113 series involving, 70 Economization of series, 411–413 Eigenfunction Bessel’s equations, 337–339 partial differential equations, 386 Eigenvalue Bessel’s equations, 337–339 definition, 57 diagonal and, 57 partial differential equations, 386 Eigenvector, 57 Elementary function, 229 Ellipsoid geometry, 24 Elliptic cylindrical coordinates, 377–378 Elliptic function definition, 230 Jacobian, 235–238 Elliptic integrals, 229–238 definitions, 229–230 series representation, 232–234 tables of values, 231–232, 234 types, 229–230 Elliptic partial differential equations, 382 Entropy conditions, 397 Error function, 242–244 derivatives, 243 integral, 243–244 relationship with normal probability distribution, 243 table of values, 241 Euler integral, 221 Euler–Macheroni constant, 3, 222 Euler–Maclaurin summation formula, 45–46 Euler numbers definitions, 38–39 list of, 39 relationships with Bernoulli numbers, 39–40 series representation, 40 Euler polynomial, 44–45 Euler’s constant, 222 Euler’s formula, 102 Euler’s method, for differential equations, 345–346 Even function definition, 115 trigonometric, 117 Exact differential equation, 325 Exponential Fourier series, 260–261 Exponential Fourier transform, 307 Exponential function derivatives, 3, 140 Euler’s formula, 102 inequalities involving, 136 integrals, 4, 7, 8, 168–171 definite, 254–255 limiting values, 136 series representation, 12, 114 Exponential integral, 168 F Factorial, asymptotic approximations to, 223, see also Gamma function False position method, for determining roots, 106–107 Faltung theorem, 297, 309 Finite difference methods, 415–417 Finite sum, 29–35 First fundamental theorem of calculus, 88 Fourier convolution, 308–309 Fourier cosine transform, 309, 311–314 Fourier series, 81–85 Bessel’s inequality, 84 bounded variation, 82 coefficients, 81, 257–265 complex form, 83 convergence, 81–82 definitions, 81–85 differentiation, 84 Dini’s condition, 82 Dirichlet condition, 82 Dirichlet expression for n’th partial sum, 83–84 Dirichlet kernel, 83 discontinuous functions, 265–268 examples, 260–268 forms of, 257–265 half-range forms, 82–83 integration, 84 Parseval identity, 84–85 Parseval relations, 257–262 periodic extension, 266 Riemann–Lebesgue lemma, 83 total variation, 82 www.TechnicalBooksPDF.com Fourier sine transform, 309, 311–312 Fourier transform, 307–314 basic properties, 308, 312 convolution operation, 308, 309 inversion integral, 308 sine and cosine transforms, 311 transform pairs, 307, 309 tables, 310–311, 313–314 Fractional order, of Bessel functions, 272–273, 277–278 Fresnel integral, 245–248 Frobenius method, for differential equations, 340–344 Frustrum, 21 Function algebraic functions, 145 beta, 224–225 complementary error, 242 error, 242 even, 115 exponential function, 114 gamma, 222, 224 hyperbolic, 134 inverse hyperbolic, 3, 12, 135 inverse trigonometric, 3, 12, 134 logarithmic, 112–113 odd, 115 periodic, 115 psi (digamma), 224 rational, 146 transcendental, 145 trigonometric, 115 Functional series, 73–74 Abel’s theorem, 74 definitions, 73–74 Dirichlet’s theorem, 74 region of convergence, 73 termwise differentiation, 74 termwise integration, 74 uniform convergence, 73 Weierstrass M test, 73 Fundamental interval, for Fourier series, 265 Fundamental theorems of calculus, 88–89 G Gamma function asymptotic representation, 223 definition, 221 graph, 225 properties, 222–224 special numerical values, 223 table of values, 226–228 427 Index Gauss divergence theorem, 366–367 Gaussian probability density function, 239–240 Gaussian quadrature, 318 General solution, of differential equations, 321, 326 Generalized L’Hôpital’s rule, 87 Generalized Parseval identity, 84–85 Generating functions, for orthogonal polynomials, 289, 294, 296, 297 Geometric figures, reference data, 13–24 Geometric series, 33, 68 Geometry, applications of definite integrals, 96–99 Gerschgorin circle theorem, 61 Gradient, vector operation, 371 Green’s first theorem, 367 Green’s second theorem, 367 Green’s theorem in the plane, 367–368 H Hadamard’s inequality, 51 Hadamard’s theorem, 51 Half angles hyperbolic identities, 125–126 trigonometric identities, 120–121 Half arguments, in Jacobian elliptic functions, 236 Half-range Fourier series, 82–83 Heat equation, 398–400 Heaviside step function, 300, 404 Hermite polynomial, 296–297 Hermitian matrix, 51 Hermitian transpose, 56 Hessian determinant, Holder’s inequality, 27–28 Homogeneous equation differential, 325–326 differential linear, 326–331 partial differential, 382 Hyperbolic functions basic relationships, 121 definite integrals, 256 definitions, 2, 121 derivatives, 141–142 graphs, 122 half-argument form, 125–126 identities, 2, 111–112, 121, 123–126 inequalities, 137 integrals, 9–10, 179–189 inverse, see Inverse hyperbolic functions multiple argument form, 124 powers, 125 series, 12, 134 sum and difference, 123 Hyperbolic partial differential equation, 382 Hyperbolic problem, 385–387 Hypergeometric equation, 344–345 I Idempotent matrix, 57 Identities complex numbers, 2, 102–103 constants, 2–3 e, Green’s theorems, 367 half angles, 120–121, 125–126 hyperbolic functions, 2, 111–112, 123–126 inverse hyperbolic, 128, 131, 132 inverse trigonometric functions, 128, 131, 132 Jacobian elliptic, 235 Lagrange’s, 27 logarithmic, 128, 131 multiple angles, 119–120, 124–125 Parseval’s, 84 trigonometric, 1, 111–112, 117–121 vector, 368 Identity matrix, 55 Ill-posed partial differential equation, 383 Imaginary part, of complex number, 26 Imaginary unity, 25 Improper integral convergence, 92–94 definitions, 92–93 divergence, 92–93 evaluation, 92–93 first kind, 92 second kind, 92 Incomplete elliptic integral, 230 Indefinite integral, see also Antiderivative algebraic functions, 5–7, 146–165 Bessel functions, 281–282 definition, 88, 364 exponential function, 167–171 hyperbolic functions, 179–189 inverse hyperbolic functions, 191–195 inverse trigonometric functions, 215–219 logarithmic function, 173–178 nonrational function, 158–165 www.TechnicalBooksPDF.com rational functions, 146–157 simplification by substitution, 197–199 trigonometric functions, 169–171, 197–199 Indicial equation, 341 Induction mathematical, 35 Inequalities absolute value integrals, 90 arithmetic–geometric, 28 Bessel’s, 84 Carleman’s, 28 Cauchy–Schwarz–Buniakowsky inequality, 27 Chebyshev, 28 comparison of integrals, 90 exponential function, 136 Hadamard’s, 51 Holder, 27 hyperbolic, 137 Jensen, 29 logarithmic, 136 Minkowski, 27 real and complex, 26–29 trigonometric, 137 Infinite products absolute convergence, 71 convergence, 71 divergence, 71 examples, 71–72 Vieta’s formula, 72 Wallis’s formula, 72 Infinite series, 68–70 Inhomogeneous differential equation ordinary, 326–327, 331–333 Initial conditions, 327 Initial point, of vector, 354 Initial value problem, 327, 382, 383 Inner product, 358–359 Integral definite, see Definite integral elliptic, see Elliptic integral of error function, 243–244 of Fourier series, 84 Fresnel, 245–246 improper, see Improper integral indefinite, see Indefinite integral inequalities, 90 inversion, 300, 308, 311 of Jacobian elliptic functions, 237, 238 line, 364–365 mean value theorem for integrals, 90 n’th repeated, 243–244 428 Index Integral (continued ) particular, see Particular integral standard, 4–11 surface, 366 volume, 368 Integral form conservation equation, 392 Taylor series, 79 Integral method, for differential equations, 342 Integrating factor, 323 Integration by parts, 88, 89–90 Cauchy principal value, 93 contiguous intervals, 89 convergence of improper type, 93–94 definitions, 88 differentiation with respect to a parameter, 90 of discontinuous functions, 85 dummy variable, 88 first fundamental theorem, 88 limits, 88–89 of matrices, 61 numerical, 315–320 of power series, 75 rational functions, 94–95 reduction formulas, 91 Romberg, 318–320 rules of, 88–90 second fundamental form, 89 substitutions, 89 trigonometric, 198–199 term by term, 74 of vector functions, 363–368 zero length interval, 85 Integration formulas, open and closed, 315–320 Interpolation methods, 409–410 Lagrange, 410 linear, 410 spline, 410–411 Inverse hyperbolic functions definitions, 3, 128, 142 derivatives, 142–143 domains of definition, 128 graphs, 130 identities, 128, 131, 132 integrals, 10–11, 191–195 principal values, 128 relationships between, 132 series, 12, 134–135 Inverse Jacobian elliptic function, 237–238 Inverse, of matrix, 56 Inverse trigonometric functions derivatives, 140–141 differentiation, 3, 128, 140 domains of definition, 128 functional relationships, 128 graphs, 129 identities, 128, 131–132 integrals, 8, 215–219 principal values, 128 relationships between, 131–132 series, 11–12, 134–135 Inversion integral Fourier transform, 308, 311 Laplace transform, 300 z-transform, 404 Irrational algebraic function, see Nonrational algebraic function Irreducible matrix, 55 Irregular point, of differential equation, 340 J Jacobian determinant, 52, 369–370 Jacobian elliptic function, 235–237 Jacobi’s theorem, 50–51 Jensen inequality, 29 K KdV equation, see Korteweg-de Vries equation KdVB equation, see Korteweg-de Vries-Burger’s equation Korteweg-de Vries-Burger’s equation, 402 Korteweg-de Vries equation, 400–401 Kronecker delta, 49 L Lagrange form, of Taylor series remainder term, 79 Lagrange’s identity, 27 Laguerre polynomials, 294–296 Laplace convolution, 301 Laplace expansion, 48 Laplace transform basic properties, 300 convolution operation, 301 definition, 299 delta function, 301 of Heaviside step function, 300 inversion integral, 300 www.TechnicalBooksPDF.com pairs, 299, 301–305 pairs, table, 301–305 z-transform and, 406–407 Laplace’s equation, 382 Laplacian partial differential equations, 383–384 vectors, 367, 371 Leading diagonal, 55 Legendre function, 289 Legendre normal form, of elliptic integrals, 230 Legendre polynomials, 289, 390 Legendre’s equation, 337 Leibnitz’s formula, 87 Length of arc by integration, 98 L’Hôpital’s rule, 87 Limit comparison test, 67 Limiting values exponential function, 136 Fresnel integrals, 246 logarithmic function, 136 trigonometric functions, 137 Line integral, 364–365 Linear constant coefficient differential equation, 327 Linear dependence, 322–323 Linear interpolation, 409 Linear second-order partial differential equation, 382 Linear superposition property, 382 Logarithm to base e, 112 Logarithmic function basic results, 112–113 as constant, 2–3 definitions, 112–113 derivatives, 3, 139–140 identities, 128, 131 inequalities involving, 136 integrals, 4, 8, 173–178 definite, 256 limiting values, 136 series, 12, 70, 126–127 Lower limit, for definite integral, 88 Lower triangular matrix, 56 M Maclaurin series Bernoulli numbers, 37 definition, 80 Mass of lamina, 99 Mathematical induction, 36–37 Matrix, 55–61 adjoint, 56 Cayley–Hamilton theory, 58 429 Index characteristic equation, 57, 58 definitions, 55–57 derivatives, 60–61 diagonal dominance, 51, 57 differentiation and integration, 60–61 eigenvalue, 57 eigenvector, 57 equality, 57 equivalent, 56 exponential, 61 Hermitian, 51 Hermitian transpose, 56 idempotent, 57 identity, 55 inverse, 56 irreducible, 55 leading diagonal, 55 lower-triangular form, 56 multiplication, 57–58 nilpotent, 57 nonnegative definite, 57 nonsingular, 52, 56 normal, 56 null, 55 orthogonal, 58 positive definite, product, 57–58 quadratic forms, 58–59 reducible, 55 scalar multiplication, 57 singular, 56 skew-symmetric, 56 square, 55 subtraction, 57 sums of powers of integers, 56 symmetric, 56 transpose, 56 Hermitian, 56 unitary, 56 Matrix exponential, 61 Maxwell’s equations, 390–391 Mean, of normal distribution, 239 Mean-value theorem for derivatives, 87 for integrals, 90 Midpoint rule, for numerical integration, 316 Minkowski’s inequality, 27 Minor, of determinant element, 48, 50 Mixed type, partial differential equation, 384 Modified Bessel functions, 274–278 Modified Bessel’s equation, 274–275, 337–338 Modified Euler’s method, for differential equations, 346 Modular angle, 230 Modulus complex number, 26, 101 elliptic integral, 230 Modulus-argument representation, 101–102 Moment of inertia, 99 Multinomial coefficient, 62 Multiple angles/arguments hyperbolic identities, 124–125 trigonometric identities, 119–120 Multiplicative inverse, of matrix, 56 Multiplicity, 105, 327 N Naperian logarithm, 112 Natural logarithm, 112 Negative, of vector, 355 Nested multiplication, in polynomials, 110 Neumann condition, for partial differential equations, 383, 384 Neumann problem, 384 Newton–Cotes formulas, 317–318 Newton–Raphson method, for determining roots, 108–111 Newton’s method, for determining roots, 108–111 Nilpotent matrix, 57 Noncommutativity, of vector product, 359 Nonhomogeneous differential equation, see Inhomogeneous differential equation Nonnegative definite matrix, 57 Nonrational algebraic functions, integrals, 158–165 Nonsingular matrix, 52, 56 Nontrivial solution, 386 Norm, of orthogonal polynomials, 285 Normal distribution, 239–242 Normal probability distribution, 240 definition, 240 relationship with error function, 243 Normalized polynomial, 285 n’th repeated integral, 243–244 n’th roots of unity, 103 Null matrix, 55 Null vector, 354 Numerical approximation, 409–417 www.TechnicalBooksPDF.com Numerical integration (quadrature) composite mid-point rule, 316 composite Simpson’s rule, 316 compsite trapezoidal rule, 316 definition, 396 Gaussian, 318 Newton–Cotes, 316 open and closed formulas, 315–316 Romberg, 318–320 Numerical methods approximation in, 409–418 Numerical methods, for differential equations, 345–352 Numerical solution of differential equations Euler’s method, 345, 346 modified Euler’s method, 346 Runge–Kutta–Fehlberg method, 348–350 Runge–Kutta method, 346, 348 two-point boundary value problem, 350–352 O Oblate spheroidal coordinates, 379 Oblique prism, 17 Odd function definition, 115 Jacobian elliptic, 235 trigonometric, 117 Open-type integration formula, 316–318 Order of determinant, 48 of differential equations, 381 Order notation, 77 Ordinary differential equations approximations in, 414–415 Bernoulli’s equation, 324 Bessel’s equation, 337–339 Cauchy–Euler type, 336–337 characteristic polynomial, 327 complementary function, 326 definitions, 321 exact, 325 general solution, 326 homogeneous, 325–331 hypergeometric, 344–345 inhomogeneous, 326–327, 330–332 Initial value problem, 327 integral method, 342 Legendre type, 337 linear, 326–336 430 Index Ordinary differential (continued ) first order, 323–324 linear dependence and independence, 322 linear homogeneous constant coefficient, 327–330 second-order, 329–330 linear inhomogeneous constant coefficient, 330–332 second-order, 333 particular integrals, 327, 330, 334–336 separation of variables, 323 singular point, 329 solution methods, 327–352 Frobenius, 340–344 Laplace transform, 328 numerical, 345–352 variation of parameters, 330–332 two-point boundary value problem, 327 Oriented surface, 367 Orthogonal coordinates, 369–379 Orthogonal matrix, 56 Orthogonal polynomials Chebyshev, 290–291 definitions, 285–286 Hermite, 296–297 Laguerre, 294–296 Legendre, 286–289 orthonormal, 285 Rodrigue’s formula, 286, 290, 295, 296 weight functions, 285 Orthogonality relations, 286, 290–291, 295, 296 P Pade approximation, 413–415 Pappus’s theorem, 24, 98 Parabolic cylindrical coordinates, 376 Parabolic partial differential equation, 382 Paraboloidal coordinates, 377 Parallelepiped, 15 Parallelogram geometry, 13 Parameter of elliptic integral, 230 Parseval formula, 405 Parseval relations, 257–265, 308, 312 Parseval’s identity, 84 Partial differential equations approximations in, 414–415 boundary value problem, 381 Burger’s equation, 400, 401 Cauchy problem, 383 characteristic curves, 392 characteristics, 392–396 classification, 381–384 conservation law, 391–392 definitions, 381–384 Dirichlet condition, 378, 384 eigenfunctions, 386 eigenvalues, 394 elliptic type, 382 hyperbolic type, 382 ill-posed problem, 383 initial boundary value problem, 382 initial value problem, 382 KdV equation, 400–402 KdVB equation, 402 Laplacian, 383 linear inhomogeneous, 382 Neumann condition, 383, 384 parabolic type, 382 physical applications, 390–392, 396–402 Poisson’s equation, 383 Rubin condition, 383, 384 separation constant, 386 separation of variables, 385–387 shock solutions, 396–398 similarity solution, 398–400 soliton solution, 402 solution methods, 385–402 systems, 390–391 Tricomi’s equation, 384 well-posed problem, 383 Partial fractions, 64–65 Partial sums, 66 Fourier series, 84 Particular integral, and ordinary differential equations definition, 327, 330 undetermined coefficients method, 334–336 Particular solution, of ordinary differential equation, 321 Pascal’s triangle, 30–31 Path, line integral along, 364 Periodic extension, of Fourier series, 266 Periodic function, 115, 117 Permutations, 62 Physical applications center of mass, 99 compressible gas flow, 391 conservation equation, 391–392 heat equation, 398–400 Maxwell’s equations, 390–391 www.TechnicalBooksPDF.com moments of inertia, 99 radius of gyration, 99 Sylvester’s law of inertia, 59 waves, 394, 396–398, 400–402 Pi constant, series, 69 Pi function, 221–222 Plane polar coordinates, 372 Poisson equation, 383–384 Polar coordinates, plane, 372 Polynomial Bernoulli’s, 43–44 characteristic, 327 Chebyshev, 290–294, 410–412 definition, 104–105 Euler, 44–45 evaluation, 110–111 Hermite, 296–297 interpolation, 409–410 Laguerre, 294–296 Legendre, 286–289 orthogonal, see Orthogonal polynomials roots, 104–111 Position vector, 361, 370 Positive definite matrix, 57 Positive definite quadratic form, 59 Positive semidefinite quadratic form, 59 Power hyperbolic functions, 125 integers, 34–35, 41–42 of series, 76 trigonometric functions, 119–120 Power series, 74–78 Cauchy–Hadamard formula, 75 circle of convergence, 74–75 definitions, 74–78 derivative, 75 error function, 242, 244 integral, 75 normal distribution, 240 product, 77 quotient, 75–76 radius of convergence, 75 remainder terms, 79 reversion, 78 standard, 11 Power series method, for differential equations, 339–340 Powers of x, integrands involving, 7, 249–251 Principle value, of complex argument, 102 431 Index Prism geometry, 17 Probability density function, 239–240 Products differentiation, 3, 87 infinite, see Infinite product matrix, 57–58 of power series, 77 types, in vector analysis, 358–361 Prolate spheroidal coordinates, 378–379 Properly posed partial differential equation, 383 Psi (digamma) function, 224 Pure initial value problem, 383 Purely imaginary number, 26 Purely real number, 25 Pyramid geometry, 15 Q Quadratic forms basic theorems, 59–60 inner product, 58 positive definite, 59 positive semi-definite, 59 signature, 59 Quadrature formula, 315 Quasilinear partial differential equation, 382 Quotient differentiation, 3, 87 of power series, 75–76 R R–K–F method, see Runge–Kutta–Fehlberg method Raabe’s test, for convergence, 67 Radius of convergence, 75 Radius of gyration, 99 Raising to a power, 76 Rankine–Hugoniot jump condition, 397 Rate of change theorem, 368 Rational algebraic functions definitions, 63, 146 integrals, 146–157 integration rules, 94–95 Real numbers, inequalities, 26–29 Real part, of complex number, 26 Rectangular Cartesian coordinates, 354–355, 372 Rectangular parallelepiped geometry, 15 Rectangular wedge geometry, 16 Reducible matrix, 55 Reduction formula, 91 Reflection formula, for gamma function, 222 Region of convergence, 73 Regula falsi method, for determining roots, 106–107 Regular point, of differential equation, 340–341 Remainder in numerical integration, 315 in Taylor series, 79 Reversion, of power series, 78 Reynolds number, 398 Rhombus geometry, 14 Riemann–Lebesgue lemma, 83 Right-handed coordinates, 354 Robin conditions, for partial differential equations, 383, 384 Robin problem, 384 Rodrigues’ formulas, 286, 290, 295, 296 Romberg integration, 318–320 Romberg method, 319–320 Roots of complex numbers, 103–104 Roots of functions, 104–111 bisection method, 105 false position method, 106–107 multiplicity, 105, 327 Newton’s method, 108–111 secant method, 106–107 Roots of unity, 103 Rouche form, of Taylor series remainder term, 79 Routh–Hurwitz conditions, for determinants, 55 Routh–Hurwitz theorem, 54–55 Runge–Kutta–Fehlberg method, for differential equations, 348–350 Runge–Kutta methods, for differential equations, 346–350 S Saltus, 82, 89 Scalar, 353 Scalar potential, 365 Scalar product, 57, 358–359 Scalar triple product, 360 Scale factor, for orthogonal coordinates, 370 Scale-similar partial differential equation, 398 Scaling, of vector, 356 Schlömilch form, of Taylor series remainder term, 79 Secant method www.TechnicalBooksPDF.com of determining roots, 108 of interpolation, 352 Second fundamental theorem of calculus, 89 Second order determinant, 47–48 Sector circular, 17–18 spherical, 22 Segment circular, 18 spherical, 23 Self-similar partial differential equation, 398 Self-similar solution, to partial differential equations, 398–400 Sense, of vector, 353 Separable variables, 323 Separation of variables method ordinary differential equation, 323 partial differential equation, 385–387 Series alternating, 68 arithmetic, 33 arithmetic–geometric, 33, 68 asymptotic, 85–86 Bernoulli numbers, 38–41 binomial, 11, 68–69 convergent, see Convergence of series differentiation of, 74 divergent, 66–67, 85 e, 70 elliptic integrals, 232–234 error function, 242, 244 Euler numbers, 40 exponential, 12, 114 Fourier, see Fourier series Fresnel integrals, 245–246 functional, 73–74 geometric, 33, 68 hyperbolic, 12, 134 infinite, 68–70 integration of, 74 inverse hyperbolic, 12, 134–135 inverse trigonometric, 11–12, 134–135 logarithmic, 12, 70, 126–127 Maclaurin, see Maclaurin series normal distribution, 240 pi, 69 power, see Power series sums with integer coefficients, 34–35, 41–42 Taylor, 79–80 telescoping, 267 trigonometric, 11–12, 133, 232–234 432 Index Series expansion Bessel functions, 270–272, 276–277 Jacobian elliptic functions, 236–237 Shock wave, 396–398, 401–402 Shooting method, for differential equations, 350–352 Signature, of quadratic form, 59 Signed complementary minor, 50 Simpson’s rules, for numerical integration, 316, 317 Sine Fourier series, 260, 264 Sine Fourier transform, 309, 311–312 Sine integrals, 245–248 Singular point, of differential equation, 340 Skew-symmetric matrix, 56 Solitary wave, 401 Solitons, 402 Solution nontrivial, 386 of ordinary differential equations, 321, 326–352 of partial differential equation, 381, 385–402 temporal behavior, 385 Sphere, 22–23 Spherical coordinates, 372–374 Spherical sector geometry, 22 Spherical segment geometry, 23 Spheroidal coordinates, 378–379 Spline interpolation, 410 Square integrable function, 85 Square matrix, 55 Steady-state form, of partial differential equation, 383 Stirling formula, 223 Stoke’s theorem, 367 Strictly convex function, 29 Sturm–Liouville equation, 388 Sturm–Liouville problem, 386, 387–389 Substitution, integration by, 89, 197–199 Subtraction matrix, 57 vector, 355–356 Sum binomial coefficients, 31–33 differentiation, 3, 86 finite, 29–35 integration, matrices, 57 powers of integers, 34–35, 41–42 vectors, 355–356 Surface area, 366 Surface integral, 366 Surface of revolution, area of, 98–99 Sylvester’s law of inertia, 59 Symmetric matrix, 56 Symmetry relation, 242 Synthetic division, 110 T Tables of values Bessel function zeros, 274 elliptic integrals, 231–232, 234 error function, 241 gamma function, 226–228 Gaussian quadrature, 319 normal distribution, 241 Taylor series Cauchy remainder, 79 definition, 79 error term in, 77 integral form of remainder, 79 Lagrange remainder, 79 Maclaurin series, 80 Rouché remainder, 79 Schlömilch remainder, 79 Taylor’s theorem, 414 Telescoping, of series, 267 Temporal behavior, of solution, 385 Terminal point, of vector, 354 Tetrahedron geometry, 16 Theorem of Pappus, 24, 98 Third-order determinant, 48 Toroidal coordinates, 375–376 Torus geometry, 24 Total variation, 82 Trace, of matrix, 56 Transcendental function, 145–146, see also Exponential function; Hyperbolic functions; Logarithmic functions; Trigonometric functions Transpose, of matrix, 56 Trapezium geometry, 13–14 Trapezoidal rule, for numerical integration, 316 Traveling wave, 394 Triangle geometry, 13 Triangle inequality, 26–27 Triangle rule, for vector addition, 355 Tricomi equation, 384 Trigonometric functions basic relationships, 117 connections with hyperbolic functions, 111–112 de Moivre’s theorem, 102–103 definitions, 115 derivatives, 3, 140 www.TechnicalBooksPDF.com differentiation, 3, 140 graphs, 116 half-angle representations, 120, 120–121 identities, 1, 111–112, 117–121 inequalities involving, 137 integrals, 4–5, 7–8, 169–171, 199–213 definite, 250–254 inverse, see Inverse trigonometric functions multiple-angle representations, 119 powers, 119–120 series, 11–12, 133, 232–234 substitution, for simplification of integrals, 197–199 sums and differences, 117, 118 Triple product, of vectors, 360–361 Two-point boundary value problem, 327, 350–352 U Undetermined coefficients oridinary differential equations, 334 partial fractions, 64–65 Uniform convergence, 73–74 Unit integer function, 403–404 Unit matrix, 55 Unit vector, 354 Unitary matrix, 56 Upper limit, for definite integral, 88 Upper triangular matrix, 56 V Vandermonde’s determinant, 52 Variance, of normal distribution, 240 Variation of parameters (constants), 330–332 Vector algebra, 355–357 components, 357–358 definitions, 353–355 derivatives, 361–363 direction cosines, 354–355 divergence theorem, 366 Green’s theorem, 367 identities, 368 integral theorems, 366 integrals, 363–368 line integral, 364 null, 354 position, 361, 370 rate of change theorem, 368 scalar product, 359–360 433 Index Stoke’s theorem, 367 subtraction, 355–356 sum, 355–356 triple product, 360–361 unit, 354 vector product, 359–361 Vector field, 366 Vector function derivatives, 361–363 integrals, 363–368 rate of change theorem, 368 Vector operator, in orthogonal coordinates, 371 Vector product, 359–361 Vector scaling, 356 Vieta’s formula, 72 Volume geometric figures, 14–24 of revolution, 96–98 Volume by integration, 96 Well-posed partial differential equation, 383 Wronskian determinant, 54, 322 test, 322–323 W Wallis’s formula, 72 Waves, 394, 396–398, 400–402 Wedge, 16, 20 Weierstrass’s M test, for convergence, 73 Weight function orthogonal polynomials, 285, 290–291, 295, 296 Z Z-transform, 403–408 bilateral, 403, 405–406 unilateral, 403, 407–408 Zero of Bessel functions, 273 of function, 104 Zero complex number, 26 www.TechnicalBooksPDF.com This Page Intentionally Left Blank www.TechnicalBooksPDF.com ...HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition www.TechnicalBooksPDF.com This Page Intentionally Left Blank www.TechnicalBooksPDF.com HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS. .. responsibility for errors must rest with the author www.TechnicalBooksPDF.com Preface to the Second Edition The publication of a second edition of my Handbook has provided me with the opportunity to correct... www.TechnicalBooksPDF.com Preface This book contains a collection of general mathematical results, formulas, and integrals that occur throughout applications of mathematics Many of the entries