Table of Integrals, Series, and Products Eighth Edition I S Gradshteyn and I M Ryzhik Daniel Zwillinger, Editor Rensselaer Polytechnic Institute, USA Victor Moll (Scientific Editor) Tulane University, Department of Mathematics Translated from Russian by Scripta Technica, Inc AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO ELSEVIER Academic Press is an imprint of Elsevier
Contents Preface to the Eighth Edition Acknowledgments The Order of Presentation of the Formulas Use of the Tables Index of Special Functions Notation Note on the Bibliographic References xvii xix xxv xxix xxxvii xli xlv 0 Introduction 1 01 Finite sums 1 011 Progressions 1 012 Sums of powers of natural numbers 1 013 Sums of reciprocals of natural numbers 3 014 Sums of products of reciprocals of natural numbers 3 015 Sums of the binomial coefficients 4 02 Numerical series and infinite products 7 021 The convergence of numerical series 7 022 Convergence tests 7 023-024 Examples of numerical series 9 025 Infinite products 15 026 Examples of infinite products 15 03 Functional series 16 030 Definitions and theorems 16 031 Power series 17 032 Fourier series 20 033 Asymptotic series 21 04 Certain formulas from differential calculus 22 041 Differentiation of a definite integral with respect to a parameter 22 042 The nth derivative of a product (Leibniz's rule) 22 043 The nth derivative of a composite function 23 044 Integration by substitution 24 1 Elementary Functions 25 11 Power of Binomials 25 111 Power series 25 112 Series of rational fractions 26 12 The Exponential Function 26 v
121 Series representation 26 122 Functional relations 27 123 Series of exponentials 27 13-14 Trigonometric and Hyperbolic Functions 28 42 130 Introduction 28 131 The basic functional relations 28 132 The representation of powers of trigonometric and hyperbolic functions in terms of functions of multiples of the argument (angle) 31 133 The representation of trigonometric and hyperbolic functions of multiples of the argument (angle) in terms of powers of these functions 33 134 Certain sums of trigonometric and hyperbolic functions 36 135 Sums of powers of trigonometric functions of multiple angles 37 136 Sums of products of trigonometric functions of multiple angles 38 137 Sums of tangents of multiple angles 39 138 Sums leading to hyperbolic tangents and cotangents 39 139 The representation of cosines and sines of multiples of the angle as finite products 40 141 The expansion of trigonometric and hyperbolic functions in power series 142 Expansion in series of simple fractions 43 143 Representation in the form of an infinite product 44 144-145 Trigonometric (Fourier) series 45 146 Series of products of exponential and trigonometric functions 50 147 Series of hyperbolic functions 51 148 Lobachevskiy's "Angle of parallelism" Yl(x) 51 149 The hyperbolic amplitude (the Gudermannian) gdx 52 15 The Logarithm 52 151 Series representation 52 152 Series of logarithms (cf 1431) 55 16 The Inverse Trigonometric and Hyperbolic Functions 55 161 The domain of definition 55 162-163 Functional relations 56 164 Series representations 60 2 Indefinite Integrals of Elementary Functions 63 20 Introduction 63 200 General remarks 63 201 The basic integrals 64 202 General formulas 65 21 Rational Functions 66 210 General integration rules 66 211-213 Forms containing the binomial a + bxk 68 214 Forms containing the binomial 1 ± xn 74 215 Forms containing pairs of binomials: a + bx and a + /3x 78 216 Forms containing the trinomial a + bxk + cx2k 78 217 Forms containing the quadratic trinomial a + bx + cx2 and powers of x 79 218 Forms containing the quadratic trinomial a + bx + cx2 and the binomial a + /3x 81 22 Algebraic functions 82 220 Introduction 82 221 Forms containing the binomial a + bxk and y/x 83
' vii 103 222-223 Forms containing ^/(a + bx)k 84 224 Forms containing \/q + bx and the binomial a + fix 88 225 Forms containing y/a + bx + cx2 92 226 Forms containing y/a + bx + cx2 and integral powers of x 94 227 Forms containing \/a + cx2 and integral powers of x 99 228 Forms containing y/a + bx + cx2 and first-and second-degree polynomials 229 integrals that can be reduced to elliptic or pseudo-elliptic integrals 104 23 The Exponential Function 106 231 Forms containing eax 106 232 The exponential combined with rational functions of x 106 24 Hyperbolic Functions 110 241-243 Powers of sinha;, cosha:, tanhx, and cothx 110 244-245 Rational functions of hyperbolic functions 125 246 Algebraic functions of hyperbolic functions 132 247 Combinations of hyperbolic functions and powers 140 248 Combinations of hyperbolic functions, exponentials, and powers 149 25-26 Trigonometric Functions 151 250 Introduction 151 251-252 Powers of trigonometric functions 152 253-254 Sines and cosines of multiple angles and of linear and more complicated func tions of the argument 161 255-256 Rational functions of the sine and cosine 171 257 Integrals containing y/a±bsinx or Va ± bcosx 179 258-262 Integrals reducible to elliptic and pseudo-elliptic integrals 184 263-265 Products of trigonometric functions and powers 214 266 Combinations of trigonometric functions and exponentials 227 267 Combinations of trigonometric and hyperbolic functions 231 27 Logarithms and Inverse-Hyperbolic Functions 237 271 The logarithm 237 272-273 Combinations of logarithms and algebraic functions 238 274 Inverse hyperbolic functions 242 275 Logarithms and exponential functions 242 28 Inverse Trigonometric Functions 242 281 Arcsines and arccosines 242 282 The arcsecant, the arccosecant, the arctangent and the arccotangent 243 283 Combinations of arcsine or arccosine and algebraic functions 244 284 Combinations of the arcsecant and arccosecant with powers of x 245 285 Combinations of the arctangent and arccotangent with algebraic functions 246 3-4 Definite Integrals of Elementary Functions 249 30 Introduction 249 301 Theorems of a general nature 249 302 Change of variable in a definite integral 250 303 General formulas 251 304 Improper integrals 253 305 The principal values of improper integrals 254 31-32 Power and Algebraic Functions 255 311 Rational functions 255
k2 k2 viii 318 312 Products of rational functions and expressions that can be reduced to square roots of first-and second-degree polynomials 256 313-317 Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with rational functions 256 318 Expressions that can be reduced to fourth roots of second-degree polynomials and their products with rational functions 315 319-323 Combinations of powers of x and powers of binomials of the form (a + fix) 324-327 Powers of x, of binomials of the form a + /3xp and of polynomials in x 324 33-34 Exponential Functions 336 331 Exponential functions 336 332-334 Exponentials of more complicated arguments 338 335 Combinations of exponentials and rational functions 342 336-337 Combinations of exponentials and algebraic functions 346 338-339 Combinations of exponentials and arbitrary powers 348 341-344 Combinations of rational functions of powers and exponentials 355 345 Combinations of powers and algebraic functions of exponentials 365 346-348 Combinations of exponentials of more complicated arguments and powers 366 35 Hyperbolic Functions 374 351 Hyperbolic functions 374 352-353 Combinations of hyperbolic functions and algebraic functions 377 354 Combinations of hyperbolic functions and exponentials 384 355-356 Combinations of hyperbolic functions, exponentials, and powers 388 36-41 Trigonometric 361 Rational functions of sines and cosines and trigonometric functions of multiple Functions 392 angles 392 362 Powers of trigonometric functions 397 363 Powers of trigonometric functions and trigonometric functions of linear functions 399 364-365 Powers and rational functions of trigonometric functions 403 366 Forms containing powers of linear functions of trigonometric functions 407 367 Square roots of expressions containing trigonometric functions 410 368 Various forms of powers of trigonometric functions 415 369-371 Trigonometric functions of more complicated arguments 419 372-374 Combinations of trigonometric and rational functions 427 375 Combinations of trigonometric and algebraic functions 438 376-377 Combinations of trigonometric functions and powers 440 378-381 Rational functions of x and of trigonometric functions 451 382-383 Powers of trigonometric functions combined with other powers 463 384 - Integrals containing \/l sin2 x, y/1 cos2 x, and similar expressions 476 385-388 Trigonometric functions of more complicated arguments combined with powers 479 389-391 Trigonometric functions and exponentials 489 392 Trigonometric functions of more complicated arguments combined with exponentials 497 393 Trigonometric and exponential functions of trigonometric functions 499 394-397 Combinations involving trigonometric functions, exponentials, and powers 501 398-399 Combinations of trigonometric and hyperbolic functions 513
ix 411-412 Combinations involving trigonometric and hyperbolic functions and powers 520 413 Combinations of trigonometric and hyperbolic functions and exponentials 526 414 Combinations of trigonometric and hyperbolic functions, exponentials, and powers 529 42-44 Logarithmic Functions 530 421 Logarithmic functions 530 422 Logarithms of more complicated arguments 533 423 Combinations of logarithms and rational functions 538 424 Combinations of logarithms and algebraic functions 541 425 Combinations of logarithms and powers 543 426-427 Combinations involving powers of the logarithm and other powers 546 428 Combinations of rational functions of In a; and powers 556 429-432 Combinations of logarithmic functions of more complicated arguments and powers 559 433-434 Combinations of logarithms and exponentials 575 435-436 Combinations of logarithms, exponentials, and powers 577 437 Combinations of logarithms and hyperbolic functions 582 438-441 Logarithms and trigonometric functions 584 442-443 Combinations of logarithms, trigonometric functions, and powers 597 444 Combinations of logarithms, trigonometric functions, and exponentials 603 45 Inverse Trigonometric Functions 603 451 Inverse trigonometric functions 603 452 Combinations of arcsines, arccosines, and powers 603 453-454 Combinations of arctangents, arccotangents, and powers 605 455 Combinations of inverse trigonometric functions and exponentials 608 456 A combination of the arctangent and a hyperbolic function 609 457 Combinations of inverse and direct trigonometric functions 609 458 A combination involving an inverse and a direct trigonometric function and a power 610 459 Combinations of inverse trigonometric functions and logarithms 610 46 Multiple Integrals 611 460 Change of variables in multiple integrals 611 461 Change of the order of integration and change of variables 612 462 Double and triple integrals with constant limits 614 463-464 Multiple integrals 617 5 Indefinite Integrals of Special Functions 623 51 Elliptic Integrals and Functions 623 511 Complete elliptic integrals 623 512 Elliptic integrals 625 513 Jacobian elliptic functions 627 514 Weierstrass elliptic functions 630 52 The Exponential Integral Function 631 521 The exponential integral function 631 522 Combinations of the exponential integral function and powers 631 523 Combinations of the exponential integral and the exponential 632 53 The Sine Integral and the Cosine Integral 632 54 The Probability Integral and Fresnel Integrals 633 55 Bessel Functions 633
X 56 Orthogonal Polynomials 634 57 Hypergeometric Functions 634 6-7 Definite Integrals of Special Functions 637 61 Elliptic Integrals and Functions 637 611 Forms containing F(x, k) 637 612 Forms containing E(x,k) 638 613 Integration of elliptic integrals with respect to the modulus 639 614-615 Complete elliptic integrals 639 616 The theta function 642 617 Generalized elliptic integrals 643 62-63 The Exponential Integral Function and Functions Generated by It 644 621 The logarithm integral 644 622-623 The exponential integral function 646 624-626 The sine integral and cosine integral functions 648 627 The hyperbolic sine integral and hyperbolic cosine integral functions 653 628-631 The probability integral 653 632 Fresnel integrals 657 64 The Gamma Function and Functions Generated by It 659 641 The gamma function 659 642 Combinations of the gamma function, the exponential, and powers 660 643 Combinations of the gamma function and trigonometric functions 663 644 The logarithm of the gamma function* 664 645 The incomplete gamma function 665 646-647 The function tp(x) 666 65-67 Bessel Functions 667 651 Bessel functions 667 652 Bessel functions combined with x and x2 672 653-654 Combinations of Bessel functions and rational functions 678 655 Combinations of Bessel functions and algebraic functions 682 656-658 Combinations of Bessel functions and powers 683 659 Combinations of powers and Bessel functions of more complicated arguments 697 661 Combinations of Bessel functions and exponentials 702 662-663 Combinations of Bessel functions, exponentials, and powers 706 664 Combinations of Bessel functions of more complicated arguments, exponentials, and powers 716 665 Combinations of Bessel and exponential functions of more complicated arguments and powers 718 666 Combinations of Bessel, hyperbolic, and exponential functions 720 667-668 Combinations of Bessel and trigonometric functions 724 669-674 Combinations of Bessel and trigonometric functions and powers 734 675 Combinations of Bessel, trigonometric, and exponential functions and powers 750 676 Combinations of Bessel, trigonometric, and hyperbolic functions 753 677 Combinations of Bessel functions and the logarithm, or arctangent 754 678 Combinations of Bessel and other special functions 755 679 Integration of Bessel functions with respect to the order 756 68 Functions Generated by Bessel Functions 760 681 Struve functions 760
xi 682 Combinations of Struve functions, exponentials, and powers 761 683 Combinations of Struve and trigonometric functions 762 684-685 Combinations of Struve and Bessel functions 763 686 Lommel functions 767 687 Thomson functions 768 69 Mathieu Functions 770 691 Mathieu functions 770 692 Combinations of Mathieu, hyperbolic, and trigonometric functions 770 693 Combinations of Mathieu and Bessel functions 774 694 Relationships between eigenfunctions of the Helmholtz equation in different coordinate systems 774 71-72 Associated Legendre Functions 776 711 Associated Legendre functions 776 712-713 Combinations of associated Legendre functions and powers 777 714 Combinations of associated Legendre functions, exponentials, and powers 715 Combinations of associated Legendre and hyperbolic functions 785 716 Combinations of associated Legendre functions, powers, and trigonometric 783 functions 786 717 A combination of an associated Legendre function and the probability integral 788 718 Combinations of associated Legendre and Bessel functions 789 719 Combinations of associated Legendre functions and functions generated by Bessel functions 794 721 Integration of associated Legendre functions with respect to the order 795 722 Combinations of Legendre polynomials, rational functions, and algebraic functions 796 723 Combinations of Legendre polynomials and powers 798 724 Combinations of Legendre polynomials and other elementary functions 799 725 Combinations of Legendre polynomials and Bessel functions 801 73-74 Orthogonal Polynomials 802 731 Combinations of Gegenbauer polynomials C (x) and powers 802 732 Combinations of Gegenbauer polynomials C%(x) and elementary functions 805 7325* Complete System of Orthogonal Step Functions 805 733 Combinations of the polynomials C^{x) and Bessel functions Integration of Gegenbauer functions with respect to the index 806 734 Combinations of Chebyshev polynomials and powers 807 735 Combinations of Chebyshev polynomials and elementary functions 809 736 Combinations of Chebyshev polynomials and Bessel functions 810 737-738 Hermite polynomials 810 739 Jacobi polynomials 814 741-742 Laguerre polynomials 816 75 Hypergeometric Functions 820 751 Combinations of hypergeometric functions and powers 820 752 Combinations of hypergeometric functions and exponentials 822 753 Hypergeometric and trigonometric functions 825 754 Combinations of hypergeometric and Bessel functions 825 76 Confluent Hypergeometric Functions 828 761 Combinations of confluent hypergeometric functions and powers 828
xii 762-763 Combinations of confluent hypergeometric functions and exponentials 830 764 Combinations of confluent hypergeometric and trigonometric functions 837 765 Combinations of confluent hypergeometric functions and Bessel functions 766 Combinations of confluent hypergeometric functions, Bessel functions, 838 and powers 839 767 Combinations of confluent hypergeometric functions, Bessel functions, exponentials, and powers 842 768 Combinations of confluent hypergeometric functions and other special 769 Integration of confluent hypergeometric functions with respect to the index functions 847 849 77 Parabolic Cylinder Functions 849 771 Parabolic cylinder functions 849 772 Combinations of parabolic cylinder functions, powers, and exponentials 850 773 Combinations of parabolic cylinder and hyperbolic functions 851 774 Combinations of parabolic cylinder and trigonometric functions 852 775 Combinations of parabolic cylinder and Bessel functions 853 776 Combinations of parabolic cylinder functions and confluent hypergeometric functions 857 777 Integration of a parabolic cylinder function with respect to the index 857 78 Meijer's and MacRobert's Functions (G and E) 858 781 Combinations of the functions G and E and the elementary functions 858 782 Combinations of the functions G and E and Bessel functions 862 783 Combinations of the functions G and E and other special functions 864 8-9 Special Functions 867 81 Elliptic Integrals and Functions 867 811 Elliptic integrals 867 812 Functional relations between elliptic integrals 872 813 Elliptic functions 874 814 Jacobian elliptic functions 875 815 Properties of Jacobian elliptic functions and functional relationships between them 879 816 The Weierstrass function p(u) 882 817 The functions (u) and <r(u) 885 818-819 Theta functions 886 82 The Exponential Integral Function and Functions Generated by It 892 821 The exponential integral function Ei(x) 892 822 The hyperbolic sine integral sbi x and the hyperbolic cosine integral chi x 895 823 The sine integral and the cosine integral: six and cix 895 824 The logarithm integral li(x) 896 825 The probability integral $(x), the Fresnel integrals S(x), C{x), the error func tion erf(x), and the complementary error function erfc(x) 896 826 Lobachevskiy's function L(x) 900 83 Euler's Integrals of the First and Second Kinds and Functions Generated by Them 901 831 The gamma function (Euler's integral of the second kind): T(z) 901 832 Representation of the gamma function as series and products 903 833 Functional relations involving the gamma function 904 834 The logarithm of the gamma function 907 835 The incomplete gamma function 908 836 The psi function ip(x) 911
xiii 837 The function 0(x) 915 838 The beta function (Euler's integral of the first kind): B(x,y) 917 839 The incomplete beta function Bx(p, q) 919 84-85 Bessel Functions and Functions Associated with Them 919 840 Definitions 919 841 Integral representations of the functions J (z) and Nv(z) 921 842 Integral representations of the functions H^\z) and H^\z) 923 843 Integral representations of the functions Iv{z) and Kv{z) 925 844 Series representation 927 845 Asymptotic expansions of Bessel functions 929 846 Bessel functions of order equal to an integer plus one-half 933 847-848 Functional relations 935 849 Differential equations leading to Bessel functions 940 851-852 Series of Bessel functions 942 853 Expansion in products of Bessel functions 948 854 The zeros of Bessel functions 950 855 Struve functions 951 856 Thomson functions and their generalizations 953 857 Lommel functions 954 858 Anger and Weber functions Jv(z) and E (z) 957 859 Neumann's and Schlafli's polynomials: On(z) and Sn(z) 958 86 Mathieu Functions 959 860 Mathieu's equation 959 861 Periodic Mathieu functions 960 862 Recursion relations for the coefficients A%^\ B^ff*' Bg 2) 960 863 Mathieu functions with a purely imaginary argument 961 864 Non-periodic solutions of Mathieu's equation 962 865 Mathieu functions for negative q 962 866 Representation of Mathieu functions as series of Bessel functions 963 867 The general theory 966 87-88 Associated Legendre Functions 967 870 Introduction 967 871 Integral representations 969 872 Asymptotic series for large values of \v\ 971 873-874 Functional relations 973 875 Special cases and particular values 977 876 Derivatives with respect to the order 978 877 Series representation 979 878 The zeros of associated Legendre functions 981 879 Series of associated Legendre functions 981 881 Associated Legendre functions with integer indices 983 882-883 Legendre functions 984 884 Conical functions 989 885 Toroidal functions 990 89 Orthogonal Polynomials 991 890 Introduction 991 891 Legendre polynomials 992
xiv 8919 Series of products of Legendre and Chebyshev polynomials 997 892 Series of Legendre polynomials 997 893 Gegenbauer polynomials C*(i) 999 894 The Chebyshev polynomials Tn(x) and Un(x) 1002 895 The Hermite polynomials Hn(x) 1005 896 Jacobi's polynomials 1007 897 The Laguerre polynomials 1009 91 Hypergeometric Functions 1014 910 Definition 1014 911 Integral representations 1014 912 Representation of elementary functions in terms of a hypergeometric functions 1015 913 Transformation formulas and the analytic continuation of functions defined by hypergeometric series 1017 914 A generalized hypergeometric series 1020 915 The hypergeometric differential equation 1020 916 Riemann's differential equation 1023 917 Representing the solutions to certain second-order differential equations using a Riemann scheme 1026 918 Hypergeometric functions of two variables 1027 919 A hypergeometric function of several variables 1031 92 Confluent Hypergeometric Functions 1031 920 Introduction 1031 921 The functions ${a,r,z) and V(a,r,z) 1032 922-923 The Whittaker functions MA,M(z) and W\,p(z) 1033 924-925 Parabolic cylinder functions Dp{z) 1037 926 Confluent hypergeometric series of two variables 1040 93 Meijer's G-Function 1041 930 Definition 1041 931 Functional relations 1043 932 A differential equation for the G-function 1044 933 Series of G-functions 1044 934 Connections with other special functions 1044 94 MacRobert's ^-Function 1045 941 Representation by means of multiple integrals 1045 942 Functional relations 1045 95 Riemann's Zeta Functions (,{z,q), and C,{z), and the Functions $(z,s,v) and 1046 951 Definition and integral representations 1046 952 Representation as a series or as an infinite product 1047 953 Functional relations 1048 954 Singular points and zeros 1049 955 The Lerch function $(z,s,v) 1049 956 The function (s) 1051 96 Bernoulli Numbers and Polynomials, Euler Numbers, the Functions v{x), u(x, a), (i(x,/3), fj,(x,0,a), X(x,y) and Euler Polynomials 1051 961 Bernoulli numbers 1051 962 Bernoulli polynomials 1052 963 Euler numbers 1054
_xv 964 The functions u(x), u{x,a), fj(x,0), (i(x,f3,a), X(x,y) 1054 965 Euler polynomials 1055 97 Constants 1056 971 Bernoulli numbers 1056 972 Euler numbers 1056 973 Euler's and Catalan's constants 1057 974 Stirling numbers 1057 10 Vector Field Theory 1061 101-108 Vectors, Vector Operators, and Integral Theorems 1061 1011 Products of vectors 1061 1012 Properties of scalar product 1061 1013 Properties of vector product 1061 1014 Differentiation of vectors 1062 1021 Operators grad, div, and curl 1062 1031 Properties of the operator V 1063 1041 Solenoidal fields 1064 1051-1061 Orthogonal curvilinear coordinates 1064 1071-1072 Vector integral theorems 1067 1081 Integral rate of change theorems 1069 11 Integral Inequalities 1071 1111 Mean Value Theorems 1071 11111 First mean value theorem 1071 11112 Second mean value theorem 1071 11113 First mean value theorem for infinite integrals 1071 11114 Second mean value theorem for infinite integrals 1072 1121 Differentiation of Definite Integral Containing a Parameter 1072 11211 Differentiation when limits are finite 1072 11212 Differentiation when a limit is infinite 1072 1131 Integral Inequalities 1072 11311 Cauchy-Schwarz-Buniakowsky inequality for integrals 1072 11312 Holder's inequality for integrals 1072 11313 Minkowski's inequality for integrals 1073 11314 Chebyshev's inequality for integrals 1073 11315 Young's inequality for integrals 1073 11316 Steffensen's inequality for integrals 1073 11317 Gram's inequality for integrals 1073 11318 Ostrowski's inequality for integrals 1074 1141 Convexity and Jensen's Inequality 1074 11411 Jensen's inequality 1074 11412 Carleman's inequality for integrals 1074 1151 Fourier Series and Related Inequalities 1074 11511 Riemann-Lebesgue lemma 1075 11512 Dirichlet lemma 1075 11513 Parseval's theorem for trigonometric Fourier series 1075 11514 Integral representation of the nth partial sum 1075 11515 Generalized Fourier series 1075
xvi 11516 Bessel's inequality for generalized Fourier series 1076 11517 Parseval's theorem for generalized Fourier series 1076 12 Fourier, Laplace, and Mellin Transforms 1077 121-124 Integral Transforms 1077 1211 Laplace transform 1077 1212 Basic properties of the Laplace transform 1077 1213 Table of Laplace transform pairs 1078 1221 Fourier transform 1087 1222 Basic properties of the Fourier transform 1088 1223 Table of Fourier transform pairs 1088 1224 Table of Fourier transform pairs for spherically symmetric functions 1090 1231 Fourier sine and cosine transforms 1091 1232 Basic properties of the Fourier sine and cosine transforms 1091 1233 Table of Fourier sine transforms 1092 1234 Table of Fourier cosine transforms 1096 1235 Relationships between transforms 1099 1241 Mellin transform 1099 1242 Basic properties of the Mellin transform 1100 1243 Table of Mellin transforms 1101 Bibliographic References 1105 Supplementary References 1109 Index of Functions and Constants 1115 Index of Concepts 1125