M. Abramowitz and I. A. Stegun.

Handbook of mathematical functions.

Table of contents :: Index to all pages and sections :: Subject index

Subject Index

A B C D E F G H I J K L M N O P Q R S T U V W Z Notations

A

Top of Index

Adam's formulas 896
Airy functions 367, 446, 510, 540, 689
- ascending series for 446
- ascending series for integrals of 447
- asymptotic expansions of 448
- asymptotic expansions of modulus and phase 449
- asymptotic forms of integrals of 449
- asymptotic forms of related functions 450
- complex zeros and associated values of Bi(z) and Bi'(z) 478
- computation of 454
- computation of zeros of 454
- definitions of 446
- differential equations 446, 448
- graphs of 446
- integral representations of 447
- integrals involving 448
- modulus and phase 449
- related functions 448
- relations between 446
- representations in terms of Bessel functions 447
- tables of 475
- tables of integrals of 478
- Wronskian for products of 448
- Wronskian relations 446
- zeros and their asymptotic expansions 450
Airy functions and their derivatives 446
- zeros and associated values of 478
Airy integrals 447
Aitken's $\delta^2$-process 18
Aitken's iteration method 879
Anger's function 498
- relation to Weber's function 498
Antilogarithm 89
Approximate values 14
Approximation methods 18
- Aitken's $\delta^2$-process 18
- Newton's method 18
- regula falsi 18
- successive substitution 18
Argument 16
Arithmetic functions 826
- table of 840
Arithmetic-geometric mean 571, 577, 578, 580, 598, 601
Arithmetic mean 10
Arithmetic progression 10
Associated Legendre functions 331
- (see Legendre functions) 332
Asymptotic expansions 15

B

Top of Index

Basic numbers 822
Bateman function 510
Bernoulli numbers 804
- table of 810
Bernoulli polynomials 803
- as sums of powers 804
- coefficients of 809
- derivatives of 804
- differences of 804
- expansions of 804
Fourier expansions of 805
- generating function for 804
- inequalities for 805
- integrals involving 805
- multiplication theorem for 804
- relations with Euler polynomials 806
- special values of 805
- symbolic operations 806
- symmetry relations 804
Bessel functions as parabolic cylinder functions 692, 697
- definite integrals 485
- modified 374, 509
- notation for 358
- of fractional order 437
- of the first kind 358, 509
- of the second kind 358, 509
- of the third kind 358, 510
- orthogonality properties of 485
- representations in terms of Airy functions 447
- spherical 437, 509
Bessel functions of half-integer order 437, 497
- zeros and associated values 467
- zeros of the derivative and associated values 468
Bessel functions, integrals 479, 485
- asymptotic expansions 480, 482
- computation of 488
- convolution type 485
- Hankel-Nicholson type 488
- involving products of 484
- polynominal approximations 481, 482
- recurrence relations 480, 483
- reduction formulas 483
- repeated 482
- simple 480
- tables of 492
- Weber-Schafheitlin type 487
Bessel functions $J_\nu(z)$, $Y_\nu(z)$ 358, 379, 381, 385
- addition theorems for 363
- analytic continuation of 361
- ascending series for 360
- asymptotic expansion for large arguments 364
- asymptotic expansions for large orders 365
- asymptotic expansions for zeros 371
- asymptotic expansions in the transition region for large orders 367
- asymptotic expansions of modulus and phase for large arguments 365
- connection with Legendre functions 362
- continued fractions for 363
- derivatives with respect to order 362
- differential equation 358
- differential equations for products 362
- formulas for derivatives 361
- generating function and associated series 361
- graphs of 359, 373
- in terms of hypergeometric functions 362
- integral representations of 360
- limiting forms for small arguments 360
- modulus and phase 365
- multiplication theorem for 363
- Neumann's expansion of an arbitrary function 363
- notation 358
- other differential equations 362
- polynomial approximations 369
- recurrence relations 361
- recurrence relations for cross-products 361
- relations between 358
- tables of 390
- uniform asymptotic expansions for large orders 358
- upper bounds 362
- Wronskian relations 360
- zeros of 370
- zeros, complex 372
- zeros, infinite products for 370
- zeros, McMahon's expansions for 371
- zeros of cross products of 374
- zeros, tables of 371, 409, 414
- zeros, uniform expansions of 371
Bessel's interpolation formula 881
Beta function 258
Biharmonic operator 885
Binary scale 1017
Binomial coefficients 10, 256, 822
- table of 10, 828
Binomial distribution 960
Binomial series 14
Binomial theorem 10
Bivariate normal probability function 936
- computation of 955
- graphs of 937
- special values of 937
Bode's rule 886

C

Top of Index

Cartesian form 16
Catalan's constant 807
Cauchy-Riemann equation 17
Cauchy's inequality 11
Characteristic function 928
Chebyshev integration 887
- abscissas for 920
Chebyshev polynomials 486, 561, 774
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 795
- graphs of 778
- values of 795
Chebyshev's inequality 11
Chi-square distribution function 958
- computation of 958
Chi-square probability function 940
- approximations to 941
- asymptotic expansion 941
- continued fraction for 941
- cumulants for 940
- non-central 942
- recurrence and differential properties of 941
- relation to the normal distribution 940
- relation to other functions 943
- series expansions for 941
- statistical properties of 943
Christoffel-Darboux formula 785
Circular functions 71, 91
- addition and subtraction of 72
- addition formulas for 72
- Chebyshev approximations 76
- continued fractions for 75
- definite integrals 78
- DeMoivre's formula 74
- differentiation formulas 77
- Euler's formulas 74
- expansion in partial fractions 75
- graphs of 72
- half-angle formulas 72
- indefinite integrals 77
- inequalities for 75
- infinite products 75
- limiting values 75
- modulus and phase 74
- multiple angle formulas 72
- negative angle formulas 72
- periodic properties of 72
- polynomial approximations 76
- products of 72
- real and imaginary parts 74
- reduction to first quadrant 73
- relations between 72
- relation to hyperbolic functions 74
- series expansions for 74
- signs of 73
- tables of 142
Circular normal distribution 936
- calculation over an offset circle 957
Clausen's integral 1005
Clebsch-Gordan coefficients 1006
Cologarithm 89
Combinatorial analysis 822
Complex numbers 16
- logarithm of 67, 90
- multiplication and division of 16
- powers of 16
- roots of 17
Confluent hypergeometric functions 262, 298, 300, 362, 377, 486, 503, 686, 691, 695, 780
- alternative notations for 504
- analytic continuation of 504
- asymptotic expansions and limiting forms 508
- Barnes-type contour integrals 506
- calculation of zeros and turning points of 513
- computation of 511
- connections with Bessel functions 506
- differential properties of 506
- expansions in series of Bessel functions 506
- general confluent equation 505
- graphing of 513
- graph of zeros of 513
- graphs of 514
- integral representations of 505
- Kummer's equation 504
- Kummer's functions 504
- Kummer's transformations 505
- recurrence relations 506
- special cases of 509
- table of 516
- table of zeros of 535
- Whittaker's equation 505
- Whittaker's functions 505
- Wronskian relations 505
- zeros and turning values 510
Conformal mapping 642
Conical functions 337
Constants
- mathematical 1
- physical, miscellaneous 5
Continued fractions 19, 22, 68, 70, 75, 81, 85, 88, 229, 258, 263, 298, 363, 932, 941, 944
Conversion factors
- mathematical 1
- physical 5
Cornish-Fisher asymptotic expansions 935
Correlation 936
Cosine integral 231, 510
- asymptotic expansions of 233
- computation of 233
- definitions 231
- graphs of 232
- integral representation of 232
- integrals involving 232
- rational approximations 233
- relation to the exponential integral 232
- series expansions for 232
- symmetry relations 232
- tables of 238, 243
Coulomb wave functions 509, 537
- asymptotic behavior of 542
- asymptotic expansions of 540
- computation of 543
- differential equation 538
- expansions in terms of Airy functions 540
- expansions in terms of Bessel-Clifford functions 539
- expansions in terms of spherical Bessel functions 540
- general solution 538
- graphs of 539, 541
- integral representations of 539
- recurrence relations 539
- series expansions for 538
- special values of 542
- tables of 546
- Wronskian relations 539
Cubic equation, solution of 17, 20
Cumulants 928
Cumulative distribution function
- multivariate 927
- univariate 927
Cunningham function 510
Cylinder functions 361

D

Top of Index

Dawson's integral 262, 298, 305, 692
- graph of 297
- table of 319
Debye functions 998
DeMoivre's theorem 74, 84
Derivatives 11
- of algebraic functions 11
- of circular functions 77
- of hyperbolic functions 85
- of inverse circular functions 82
- of inverse hyperbolic functions 88
- of logarithmic functions 69
- partial 883
Differences 877
- central 877
- divided 877
- forward 877
- in terms of derivatives 883
- mean 877
- reciprocal 878
Differential equations 896
- of second order with turning points 450
- ordinary first order 896
- solution by Adam's formulas 896
  - by Gill's method 896
  - by Milne's method 896
  - by point-slope formula 896
  - by predictor-corrector methods 896
  - by Runge-Kutta method 896
- solution by trapezoidal formula 896
- system of 897
Differentiation 882
- Everett's formula 883
- Lagrange's formula 882
- Markoff's formula 883
Differentiation coefficients 882
- table of 914
Digamma function 258
- (see psi function)
Dilogarithm function 1004
Distribution functions 927
- asymptotic expansions of 935
- characteristics of 928
- continuous 927
- discrete 927
- inequalities for 931
- lattice 927
- one-dimensional continuous 930
- one-dimensional discrete 929
Divisor functions 827
- table of 840
Double precision operations 21

E

Top of Index

Economization of series 791
Edgeworth asymptotic expansion 935, 955
Einstein functions 999
Elliptic functions
- Jacobian (see Jacobian elliptic functions) 567
- Weierstrass (see Weierstrass elliptic functions) 627
Elliptic integrals 587
- amplitude 590
- canonical forms 589
- characteristic 590
- definition 589
- graphs of the complete 592
- graphs of the first kind 592, 593
- graphs of the incomplete 593, 594
- graphs of the second kind 592, 594
- graphs of the third kind 600
- modular angle 590
- modulus 590
- of the first kind 589
- of the second kind 589
- of the third kind 590, 599
- parameter 590, 602
- reduction formulas 589, 597
- reduction to canonical form 600
- relation to Weierstrass elliptic functions 649
- tables of complete 608
- tables of the incomplete 613
- tables of the third kind 625
Elliptic integrals, complete 590
- computation of 601
- infinite series for 591
- Legendre's relation 591
- limiting values 591
- of the first kind 590
- of the second kind 590
- of the third kind 599, 605
- polynomial approximations 591
- q-series for 591
- relation to hypergeometric functions 591
Elliptic integrals, incomplete 592
- amplitude of any magnitude 592
- amplitude near $\pi/2$ 593
- complex amplitude 592
- computations involving 595, 602, 605
- imaginary amplitude 594
- Jacobi's imaginary transformation 592
- negative amplitude 592
- negative parameter 593
- numerical evaluation of 595
- parameter greater than unity 593
- special cases of 594
Elliptical coordinates 752
Equianharmonic case 652
Error
- absolute 14
- percentage 14
- relative 14
Error function 262, 297, 301, 304, 306, 509
- altitude chart in the complex plane 298
- asymptotic expansion of 298
- complex zeros of 329
- continued fraction for 298
- definite and indefinite integrals related to 302
- derivatives of 298
- graphs of 297
- inequalities for 298
- infinite series approximation for the complex function 299
- integral representation of 297
- rational approximations 299
- relation to the confluent hypergeometric function 298
- repeated integrals of 299
- series expansions for 297
- symmetry relations 297
- table for complex arguments 325
- table of repeated integrals of 317
- tables of 310, 312, 316
- value at infinity 298
Eta functions 577
Euler function 826
- table of 840
Euler-Maclaurin formulas 806
Euler-Maclaurin summation formula 16, 22, 806, 886
Euler numbers 804
- table of 810
Euler polynomials 803
- as sums of powers 804
- coefficients of 809
- derivatives of 804
- differences of 804
- expansions of 804
- Fourier expansions of 805
- generating function for 804
- inequalities for 805
- integrals involving 805
- multiplication theorem for 804
- relations with Bernoulli polynomials 806
- special values of 805
- symbolic operations 806
- symmetry relations 804
Euler summation formula 806
Euler-Totient function 826
Euler's constant 255
Euler's formula 74, 255
Euler's integral 255
Euler's transformation of series 16, 21
Everett interpolation coefficients 880
- relation to Lagrange coefficients 880
Everett's formula 880, 883
Excess 928
Expected value operator 928
Exponential function 69, 90, 509
- Chebyshev approximation 71
- continued fractions for 70
- differentiation formulas 71
- Euler's formula 74
- graph of 70
- identities 70
- indefinite integrals 71
- inequalities for 70
- limiting values 70
- periodic property of 70
- polynomial approximations 71
- series expansions for 69
- tables of 116, 140, 219
Exponential integral 227, 262, 510
- asymptotic expansion of 231
- computation of 233
- continued fraction for 229
- definite integrals 230
- derivatives of 230
- graphs of 228
- indefinite integrals 230
- inequalities for 229
- interrelations 228
- polynomial approximations 231
- rational approximations 231
- recurrence relations 229
- relation to incomplete gamma function 230
- relation to spherical Bessel functions 230
- series expansions for 229
- tables of 238, 243, 245, 248, 249, 251

F

Top of Index

F-distribution function 946
- approximations to 947, 948
- computation of 961
- limiting forms 947, 948
- non-central 947
- reflexive relation 946
- relation to the $\chi^2$-distribution 946
- relation to Students t-distribution 947
- series expansions for 946, 948
- statistical properties of 946
Factorial function 255
- (see gamma function)
Factorization
- table of 844, 864
Faltung theorem 1020
Filon's quadrature formula 890
- coefficients for 924
Floquet's theorem 727
Fresnel integrals 262, 300, 440
- asymptotic expansions of 302
- auxiliary functions 300, 323
- complex zeros of 329
- definition 300
- derivatives of 301
- graph of 301
- integrals involving 303
- interrelations 300
- maxima and minima of 329
- rational approximations 302
- relation to spherical Bessel functions 301
- relation to the confluent hypergeometric function 301
- relation to the error function 301
- series expansion for 301
- symmetry relations 301
- table of 321, 323
- value at infinity 301
Fundamental period parallelogram 629

G

Top of Index

Gamma function 255, 263
- asymptotic formulas 257
- binomial coefficient 256
- continued fraction for 258
- definite integrals 258
- duplication formula 256
- Euler's formula 255
- Euler's infinite product 255
- Euler's integral 255
- fractional values of 255
- Gauss' multiplication formula 256
- graph of 255
- Hankers contour integral 255
- in the complex plane 256
- integer values of 255
- Pochhammer's symbol 256
- polynomial approximations 257
- power series for 256
- recurrence formulas 256
- reflection formula 256
- series expansion for $1/\Gamma(z)$ 256
- Stirling's formula 257
- tables of 267, 272, 274, 276
- triplication formulas 256
- Wallis' formula 258
Gauss series 556
Gaussian integration 887
- abscissas and weight factors for 916
- for integrands with a logarithmic singularity 920
- of moments 921
Gaussian probability function 931
Gauss' transformation 573
Gegenbauer polynomials 561
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 794
- graphs of 776
Generalized hypergeometric function 362, 377, 556
Generalized Laguerre polynomials 771
- (see orthogonal polynomials)
Generalized mean 10
Geometric mean 10
Geometric progression 10
Gill's method 896
Gudermannian 77

H

Top of Index

Hankel functions 358, 379, 510
Hankel's contour integral 255
Harmonic analysis202, 881
Harmonic mean 10
Haversine 78
Heaviside expansion theorem 1021
Hermite functions 509, 691
Hermite integration 890
- abscissas and weight factors for 924
Hermite polynomials 300, 510, 691, 775
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 801
- graph of 780
- values of 802
Heuman's lambda function 595
- graph of 595
- table of 622
Hh function 300, 691
Hölder's inequality for integrals 11
- for sums 11
Homer's scheme 788
Hyperbolic functions 83
- addition and subtraction of 84
- addition formulas for 83
- continued fraction for 85
- DeMoivre's theorem 84
- differentiation formulas 85
- graph of 83
- half-angle formulas 83
- indefinite integrals 86
- infinite products 85
- modulus and phase 84
- multiple angle formulas 84
- negative angle formulas 83
- periodic properties of 83
- products of 84
- real and imaginary parts 84
- relations between 83
- relation to circular functions 83
- series expansions for 85
- tables of 213, 219
Hypergeometric differential equation 562
- solution of 563
Hypergeometric functions 332, 335, 336, 362, 377, 487, 555, 779
- as Legendre functions 561
- as polynomials 561
- as reductions of Riemann's P-function 565
- asymptotic expansions of 565
- differentiation formulas 557
- Gauss series 556
- Gauss' relations for contiguous functions 557
- integral representations 558
- special cases of 561
- special elementary cases 556
- special values of the argument 556
- transformation formulas 559

I

Top of Index

Incomplete beta function 263, 944
- approximations to 945
- asymptotic expansions of 945
- computation of 959
- continued fraction for 944
- recurrence formulas 263, 944
- relation to other functions 945
- relation to the binomial expansion 263
- relation to the $\chi^2$-distribution 944
- relation to the hypergeometric function 263
- series expansion for 944
- symmetry relation 263
Incomplete gamma function 230, 260, 486, 509
- as a confluent hypergeometric function 262
- asymptotic expansions of 263
- computation of 959
- continued fraction for 263
- definite integrals 263
- derivatives and differential equations 262
- graph of 261
- Pearson's form of 262
- recurrence formulas 262
- series developments for 262
- special values of 262
- table of 978
Indeterminate forms (L'Hospital's rule) 13
Inequality
- Cauchy's 11
- Chebyshev's 11
- Hölder's for integrals 11
- Hölder's for sums 11
- Minkowski's for integrals 11
- Minkowski's for sums 11
- Schwarz's 11
- triangle 11
Integral of a bivariate normal distribution over a polygon 956
Integrals
- of circular functions 77
- of exponential functions 71
- of hyperbolic functions 86
- of inverse circular functions 82
- of inverse hyperbolic functions 88
- of irrational algebraic functions 12
- of logarithmic functions 69
- of rational algebraic functions 12
Integration 885
- Bode's rule 886
- by parts 12
- Chebyshev's equal weight formula 887
- Euler-Maclaurin summation formula 886
- Filon's formula 890
- five-point rule for analytic functions 887
- Gaussian type formulas 887
- iterated integrals 891
- Lagrange formula 886
- Lobatto's integration formula 888
- multidimensional 891
- Newton-Cotes formula 886
- Radau's integration formula 888
- Simpson's rule 886
- trapezoidal rule 885
Interpolation 878
- Aitken's iteration method 879
- Bessel's formula 881
- bivariate 882
- Everett's formula 880
- harmonic analysis 881
- inverse 881
- Lagrange formula 878
- Newton's divided difference formula 880
- Newton's forward difference formula 880
- Taylor's expansion 880
- Thiele's formula 881
- throwback formulas 880
- trigonometric 881
Invariants 629
- tables of 680
Inverse circular functions 22, 79, 92
- addition and subtraction of 80
- Chebyshev approximations 82
- continued fractions for 81
- differentiation formulas 82
- graph of 79
- indefinite integrals 82
- logarithmic representation 80
- negative arguments 80
- polynomial approximations 81
- real and imaginary parts 80
- relation to inverse hyperbolic functions 80
- series expansions for 81
- table of 203
Inverse hyperbolic functions 86, 93
- addition and subtraction of 87
- continued fractions for 88
- differentiation formulas 88
- graphs of 86
- indefinite integrals 88
- logarithmic representations 87
- negative arguments 87
- relation to inverse circular functions 87
- series expansions for 88
- tables of 221

J

Top of Index

Jacobian elliptic functions 567
- addition theorems for 574
- approximations in terms of circular functions 573
- approximations in terms of hyperbolic functions 574
- calculation by use of the arithmetic-geometric mean 571
- calculation of 579, 581
- change of argument 572
- change of parameter 573
- classification of 570
- complex arguments 575
- definitions 569
- derivatives of 574
- double arguments 574
- graphs of 570
- half-arguments 574
- integrals 575
- integrals of the squares 576
- Jacobi's imaginary transformation 574
- Landen transformation 573
- leading terms of series in powers of u 575
- parameter 569
- principal terms 572
- quarter periods 569
- reciprocal parameter 573
- relations between squares of the functions 573
- relation to the copolar trio 570
- relation with Weierstrass functions 649
- series expansions in terms of the nome q 575
- special arguments 571
Jacobi's eta function 577
Jacobi's polynomials 561, 773
- (see orthogonal polynomials)
- coefficients for 793
- graphs of 773, 776
Jacobi's theta function (see theta functions) 576
Jacobi's zeta function 578, 595
- addition theorem for 595
- calculation by use of the arithmetic-geometric mean 578
- graph of 595
- Jacobi's imaginary transformation 595
- q-series for 595
- relation to theta functions 578, 595
- special values 595
- table of 619

K

Top of Index

Kelvin functions 379, 387, 509
- ascending series for 379
- ascending series for products of 381
- asymptotic expansions for large arguments 381
- asymptotic expansions for large zeros 383
- asymptotic expansions of modulus and phase 383
- asymptotic expansions of products 383
- definitions 379
- differential equations 379
- expansion's in series of Bessel functions 381
- graphs of 382
- indefinite integrals 380
- modulus and phase 382
- of negative argument 380
- polynomial approximations 384
- recurrence relations 380
- recurrence relations for products 380
- relations between 379
- tables of 430
- uniform asymptotic expansions for large orders 384
- zeros of functions of order zero 381
Kronecker delta 822
Kummer functions 504
Kummer's transformation of series 16

L

Top of Index

Lagrange differentiation formula 882
Lagrange integration coefficients 886
- table of 915
Lagrange interpolation coefficients 878
- table of 900
Lagrange interpolation formula 878
Lagrange's expansion 14
Laguerre integration 890
- abscissas and weight factors for 923
Laguerre polynomials 509, 510, 773
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 799
- graph of 780
- values of 800
Lamé's equation 641
Landen's transformation
- ascending 573, 598, 604
- descending 573, 597, 604
Laplace transforms 1019
- definition 1020
- operations 1020
- tables of 1021
Laplace-Stieltjes transforms 1029
- tables of 1029
Laplace's equation 17
Laplacian 885
- in spherical coordinates 752
Laurent series 635
Least square approximations 790
Legendre functions 332, 362, 377, 561, 774
- asymptotic expansions of 335
- computation of 339
- explicit expressions for 333
- graphs of 338, 780
- integral representations of 335
- integrals involving 337
- notation for 332
- of negative argument 333
- of negative degree 333
- of negative order 333
- recurrence relations 333
- relation to elliptic integrals 337
- relations between 333
- Rodrigues' formula 334
- special values of 334
- summation formulas 335
- tables of 342
- trigonometric expansions of 335
- values on the cut 333
- Wronskian relation 333
Legendre polynomials 332, 486, 773
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 798
- graph of 338, 780
- values of 342
Legendre's differential equation 332
- solutions of 332
Leibniz' theorem for
- differentiation of a product 12
- differentiation of an integral 11
Lemniscate constant 658
Lemniscatic case 658
L'Hospital's rule 13
Lobatto integration 888
- abscissas and weight factors for 920
Logarithmic function 67, 89
- change of base 67, 89
- characteristic 89
- Chebyshev approximations 69
- common 68, 89
- continued fractions for 68
- definite integrals 69
- differentiation formula 69
- graph of 67
- indefinite integrals 69
- inequalities for 68
- limiting values 68
- mantissa 89
- polynomial approximations 68
- series expansions for 68
- tables of 95
Logarithmic integral 228, 231, 510
- graph of 231

M

Top of Index

Markoff's differentiation formula 883
Mathematical constants 1
- in octal scale 1017
Mathieu functions 721
- asymptotic representations of 740
- comparative notation for 744
- expansions for small 5, 730
- expansions in terms of parabolic cylinder functions 742
- graphs of 725, 734
- integral equations for 735
- integral representations of 736
- joining factors, table of 748
- normalization of 732
- orthogonality properties of 732
- other properties of 735, 738
- power series in q for periodic solutions 725
- proportionality factors 735
- recurrence relations among the coefficients 723
- special cases of 728
- special values of 740
- table of coefficients for 750
- table of critical values of 748
- zeros of 739
Mathieu's equation 722
- characteristic exponent 727
  - generation of 727
  - graphs of 728
- characteristic values 722, 748
  - asymptotic expansions of 726
  - determination of 722
  - graph of 724
  - power series for 724
- Floquet solutions 727
- other solutions 730
- relation to spheroidal wave equation 722
- solutions involving products of Bessel functions 731
- stability regions of 728
Mathieu's modified equation 722
- radial solutions of 732
Maxima 14
Mean 928
- arithmetic 10
- generalized 10
- geometric 10
- harmonic 10
Milne's method 896
Minima 14
Minkowski's inequality
- for integrals 11
- for sums 11
Miscellaneous functions 997
Möbius function 826
Modified Bessel functions, $I_\nu(z)$, $K_\nu(z)$ 374
- analytic continuation of 376
- ascending series for 375
- asymptotic expansions for large arguments 377
- connection with Legendre functions 377
- derivatives with respect to order 377
- differential equation 374
- formulas for derivatives 376
- generating function and associated series 376
- graphs of 374
- in terms of hypergeometric functions 377
- integral representations of 376
- limiting forms for small arguments 375
- multiplication theorems for 377
- Neumann series for $K_\nu(z)$ 377
- other differential equations 377
- polynomial approximations 378
- recurrence relations 376
- relations between 375
- tables of 416
- uniform asymptotic expansions for large orders 378
- Wronskian relations 375
- zeros of 377
Modified Mathieu functions 722
- graphs of 734
Modified spherical Bessel functions 443, 453, 498
- addition theorems for 445
- ascending series for 443
- computation of 335
- definitions 443
- degenerate forms 445
- derivatives with respect to order 445
- differential equation 443
- differentiation formulas 444
- duplication formula 445
- formulas of Rayleigh's type 445
- generating functions for 445
- graphs of 444
- recurrence relations 444
- representations by elementary functions 443
- tables of 469
- Wronskian relations 443
Modified Struve functions 498
- asymptotic expansion for large z 498
- computation of 499
- graph of 498
- integral representations of 498
- integrals 498
- power series expansion for 498
- recurrence relations 498
- relation to modified spherical Bessel functions 498
- tables of 501
Modulus 16
Moments 928
Multidimensional integration 891
Multinomial coefficients 823
- table of 831

N

Top of Index

Neumann's polynomial 363
Neville's notation 578
Neville's theta functions 578
- expression as infinite products 579
- expression as infinite series 579
- graphs of 578
- tables of 582
Newton coefficients 880
- relation to Lagrange coefficients 880
Newton interpolation formula 880, 883
Newton's method of approximation 18
Newton-Cotes' formula 886
Nome 591, 602
- table of 608, 610, 612
Normal probability density function, derivatives of 933
Normal probability function 931
- asymptotic expansions of 932
- bounds for 933
- computation of 953
- continued fraction for 932
- error curves for 933
- polynomial and rational approximations 932
- power series for 932
- relation to other functions 934
- values of x for extreme values of P(x) and Q(x) 977
- values of x in terms of P(x) and Q(x) 976
- values of Z(x) in terms of P(x) and Q(x) 975
Normal probability functions and derivatives 933
- tables of 966
Normal probability integral
- repeated integrals of 934
Number theoretic functions 826

O

Top of Index

Oblate spheroidal coordinates 752
Octal scale 1017
Octal tables 1017
Operations with series 15
Orthogonal polynomials 771
- as confluent hypergeometric functions 780
- as hypergeometric functions 779
- as Legendre functions 780
- as parabolic cylinder functions 780
- change of interval of orthogonality 790
- coefficients for 793
- definition 773
- differential equations 773, 781
- differential relations 783
- evaluation of 788
- explicit expressions for 775
- generating functions for 783
- graphs of 773, 776-780
- inequalities for 786
- integral representations of 784
- integrals involving 785
- interrelations 777
- limit relations 787
- of a discrete variable 788
- orthogonality relations 774
- powers of x in terms of 793, 794- 801
- recurrence relations, miscellaneous 773, 782
- recurrence relations with respect to degree n 782
- Rodrigues' formula 773, 785
- special values of 777
- sum formulas 785
- tables of 795, 796, 800, 802
- zeros of 787

P

Top of Index

Parabolic cylinder functions $U(a,x)$, $V(a,x)$ 300, 509, 685, 780
- asymptotic expansions of 689
- computation of 697
- connection with Bessel functions 692, 697
- connection with confluent hypergeometric functions 691
- connection with Hermite polynomials and functions 691
- connection with probability integrals and Dawson's integral 691
- Darwin's expansions 689
- differential equation 686
- expansions for a large, x moderate 689
- expansions for x large, a moderate 689
- expansions in terms of Airy functions 689
- integral representations of 687
- modulus and phase 690
- power series in x for 686
- recurrence relations 688
- standard solutions 687
- table of 700
- Wronskian and other relations 687
- zeros of 696
Parabolic cylinder functions $W(a,x)$ 692
- asymptotic expansions of 693
- complex solutions 693
- computation of 699
- connection with Bessel functions 695
- connection with confluent hypergeometric functions 695
- Darwin's expansions 694
- differential equation 686, 692
- expansions for a large, x moderate 694
- expansions for x large, a moderate 693
- expression in terms of Airy functions 693
- integral representations of 693
- modulus and phase 695
- power series in x for 692
- standard solutions 692
- table of 712
- Wronskian and other relations 693
- zeros of 696
Parameter m 569, 602
- table of 612
Partitions 825
- into distinct parts 825
- tables of 831, 836
- unrestricted 825
Pearson's form of the incomplete gamma function 262
Pentagamma function 260
- (see polygamma functions)
Percentage points of the $\chi^2$-distribution
- values of $\chi^2$ in terms of Q and $\nu$ 984
Percentage points of the F-distribution
- values of F in terms of Q, $\nu_1$, $\nu_2$ 986
Percentage points of the t-distribution
- values of t in terms of A and $\nu$ 990
Planck's radiation function 999
Plane triangles, solution of 78, 92
Pochhammer's symbol 256
Point-slope formula 896
Poisson-Charlier function 509
Poisson distribution 959
- table of cumulative sums of 978
Polar form 16
Polygamma functions 260
- asymptotic formulas 260
- fractional values of 260
- integer values of 260
- multiplication formula for 260
- recurrence formula 260
- reflection formula 260
- series expansions for 260
- tables of 267, 271
Polynomial evaluation 788
Powers
- computation of 19, 90
- general 69
- graph of 19
- of complex numbers 16
- of two 1016
- of xⁿ/n! 818
- tables of 24
Predictor-corrector method 896
Primes 231
- table of 870
Primitive roots 827
- table of 864
Probability density function 931
- asymptotic expansion of 935
Probability functions 927
Probability integral 262, 691
- of the $\chi^2$-distribution, table of 978
Progressions arithmetic 10
- geometric 10
Prolate spheroidal coordinates 752
Pseudo-lemniscatic case 662
Psi function 258, 264
- asymptotic formulas 259
- definite integrals 259
- duplication formula 259
- fractional values of 258
- graph of 258
- in the complex plane 259
- integer values of 258
- recurrence formulas 258
- reflection formula 259
- series expansions for 259
- tables of 267, 272, 276, 288
- zeros of 259

Q

Top of Index

Quadratic equation, solution of 17, 19
Quartic equation, resolution into quadratic factors 17, 20

R

Top of Index

Radau's integration formula 888
Random deviates, generation of 949
Random numbers 949
- methods of generation of 949
- table of 991
Repeated integrals of the error function 299
- as a single integral 299
- asymptotic expansion of 300
- definition 299
- derivatives of 300
- differential equation 299
- graph of 300
- power series for 299
- recurrence relations 299
- relation to Hermite polynomials 300
- relation to parabolic cylinder functions 300
- relation to the confluent hypergeometric function 300
- relation to the Hh function 300
- table of 317
- value at zero 300
Representation of numbers 1012
Reversion of series 16, 882
Riccati-Bessel functions 445
- definitions 445
- differential equation 445
- Wronskian relations 445
Riemann zeta function 256, 807
- special values of 807
Riemann's differential equation 564
- solutions of 564
Riemann's P function 564
- transformation formulas 565
Ring functions 336
Rodrigues' formula 334, 773, 785
Roots
- computation of 19, 89
- graph of 19
- of complex numbers 17, 20
- tables of 24, 223
Runge-Kutta methods 896

S

Top of Index

Scales of notation 1011
- general conversion methods 1012
Schwarz's inequality 11
Sectoral harmonics 332
Series 14
- binomial 14
- Euler-Maclaurin summation formula 16, 22
- Euler's transformation of 16, 21
- exponential 69
- Kummer's transformation of 16
- Lagrange's expansion 14
- logarithmic 68
- operations with 15
- reversion of 16
- Taylor's 14
- trigonometric 74
Shifted Chebyshev polynomials 774
- (see orthogonal polynomials)
Shifted Legendre polynomials 774
- (see orthogonal polynomials)
Sievert integral 1000
Simpson's rule 886
Sine integral 231, 510
- asymptotic expansions of 233
- computation of 233
- definitions 231
- graphs of 232
- integral representation of 232
- integrals 232
- rational approximations 233
- relation to exponential integral 232
- series expansions for 232
- symmetry relations 232
- tables of 238, 243
Skewness 928
Spence's integral 1004
Spherical Bessel functions 230, 301, 435, 540
- addition theorems for 440
- analytic continuation of 439
- ascending series for 437
- complex zeros of $h_n^{(1)}(z)$, $h_n^{(2)}(z)$ 441
- computation of 452
- cross products of 439
- definitions 437
- degenerate forms 440
- derivatives with respect to order 440
- differential equation 437
- differentiation formulas 439
- duplication formula 440
- Gegenbauer's generalization for 438
- generating functions for 439
- graphs of 438
- infinite series involving $j_n^2(z)$ 440
- limiting values as $z \to 0$ 437
- modulus and phase 439
- Poisson's integral for 438
- Rayleigh's formula for 439
- recurrence relations 439
- relation to Fresnel integrals 440
- representations by elementary functions 437
- tables of 457
- Wronskian relations 437
- zeros and their asymptotic expansions 440
Spherical polynomials (Legendre) 332
- (see orthogonal polynomials)
Spherical triangles, solution of 79
Spheroidal wave functions 751
- asymptotic behavior of 756
- asymptotic expansions of 755
- characteristic values for 753, 756
- differential equations 753
- evaluation of coefficients for 755
- expansions for 755
- joining factors for 757
- normalization of 755
- tables of 766
- table of eigenvalues of 760
- table of prolate joining factors 769
Stirling numbers 824
- table of the first kind 833
- table of the second kind 835
Stirling's formula 257
Struve's functions 495
- asymptotic expansions for large orders 498
- asymptotic expansions for large |z| 497, 498
- computation of 499
- differential equation 496
- graphs of 496
- integral representations of 496, 498
- integrals 497, 498
- modified 498
- power series expansion for 496, 498
- recurrence relations 496, 498
- relation to Weber's function 498
- special properties of 497
- tables of 501
Student's t-distribution 948
- approximations to 949
- asymptotic expansion of 949
- limiting distribution 949
- non-central 949
- series expansions for 948
- statistical properties of 948
Subtabulation 881
Summable series 1005
Summation of rational series 264
Sums of positive powers 813
Sums of powers 804
Sums of reciprocal powers 807, 811
Systems of differential equations of first order 897

T

Top of Index

Taylor expansion 880
Taylor's formula 14
Tesseral harmonics 332
Tetrachoric functions 934
Tetragamma function 260
- (see polygamma functions)
Theta functions 576
- addition of quarter-periods 577
- calculation by use of the arithmetic-geometric mean 577, 580
- expansions in terms of the nome q 576
Jacobi's notation for 577
- logarithmic derivatives of 576
- logarithms of sum and difference 577
Neville's notation for 578, 582
- relations between squares of the functions 576
- relation to Jacobi's zeta function 578
- relation with Weierstrass elliptic functions 650
Thiele's interpolation formula 881
Toroidal functions 336
Toronto function 509
Trapezoidal rule 885
Triangle inequality 11
Trigamma function 260
- (see polygamma functions)
Trigonometric functions 71
- (see circular functions)
Truncated exponential function 70, 262

U

Top of Index

Ultraspherical polynomials 774
- (see orthogonal polynomials)
- coefficients for and xⁿ in terms of 794
- graphs of 776
Unit step function 1020

V

Top of Index

Variance 928
Variance-ratio distribution function 946
- (see F-distribution function)
Vector-addition coefficients 1006

W

Top of Index

Wallis' formula 258
Wave equation
- in prolate and oblate spheroidal coordinates 752
Weber's function 498
- relation to Anger's function 498
- relation to Struve's function 498
Weierstrass elliptic functions 627
- addition formulas for 635
- case $\Lambda = 0$ 651
- computation of 663
- conformal mapping of 642, 654, 659
- definitions 629
- derivatives of 640
- determination of periods from given invariants 665
- determination of values at half-periods, etc., from given periods 664
- differential equation 629, 640
- discriminant 629
- equianharmonic case 652
- expressing any elliptic function in terms of $\cal{P}$ and $\cal{P}^\prime$ 651
- fundamental period parallelogram 629
- fundamental rectangle 630
- homogeneity relations 631
- integrals 641
- invariants 629
- Legendre's relation 634
- lemniscatic case 658
- maps of 642, 654, 659
- multiplication formulas 635
- other series involving $\cal{P}$, $\cal{P}^\prime$, $\zeta$ 639
- pseudo-lemniscatic case 662
- reduction formulas 631
- relation with complete elliptic integrals 649
- relations with Jacobi's elliptic functions 649
- relations with theta functions 650
- reversed series for $|\cal{P}|$, $|\cal{P}^\prime|$, $|\zeta|$ 638
- reversed series for small $|\sigma|$ 640
- series expansions for 635
- special values and relations 633
- symbolism 629
- tables of 673
Whittaker functions 505
Wigner coefficients 1006

Z

Top of Index

Zeta function
- Jacobi's 578
- Riemann's 256, 807

Index of Notations

Top of Index

$(a)_n = \Gamma(a+n)/\Gamma(a)$ (Pochhammer's symbol) 256
$a_r(q)$ characteristic value of Mathieu's equation 722
$A(x) =2P(x)-1$ normal probability function 931
Ai(z) Airy function 446
A.G.M. arithmetic-geometric mean 571
am z amplitude of the complex number z 16
antilog antilogarithm (log^-1) 89
arcsin z, arccos z, arctan z, arccot z, arcsec z, arccsc z inverse circular functions 79
arcsinh z, arccosh z, arctanh z, arccoth z, arcsech z, arccsch z inverse hyperbolic functions 86
arg z argument of z 16
$b_r(q)$ characteristic value of Mathieu's equation 722
$B_n$ Bernoulli number 804
$B_n(x)$ Bernoulli polynomial 804
$ber_\nu x$, $bei_\nu x$, Kelvin functions 379
Bi(z) Airy function 446
cd, cs, cn, Jacobian elliptic functions 570
c.d.f. cumulative distribution function 927
$ce_r(z,q)$ Mathieu function 725
Cn, Dn, Sn integrals of the squares of Jacobian elliptic functions 576
$C(x)$ Fresnel integral 300
$C_n(x)$ Chebyshev polynomial of the second kind 774
$C(x,a)$ generalized Fresnel integral 262
$Ce_r(z,q)$ modified Mathieu function 732
$C_1(z)$, $C_2(z)$ Fresnel integrals 300
$C_n^{(\alpha)}(x)$ ultraspherical (Gegenbauer) polynomial 774
Chi(z) hyperbolic cosine integral 231
Ci(z) cosine integral 231
Cin(z) cosine integral 231
Cinh(z) hyperbolic cosine integral 231
colog cologarithm 89
covers A, coversine A 78
dc, ds, Jacobian elliptic functions 570
$\mathrm{dn}=\Delta(\phi)$ delta amplitude (Jacobian elliptic function) 569
$D_\nu(x)$ parabolic cylinder function (Whittaker's form) 687
$e_1$, $e_2$, $e_3$, roots of a polynomial (Weierstrass form) 629
$e^z$ exponential function 69
$e_n(z)$ truncated exponential function 262
E($\phi$ \a) elliptic integral of the second kind 589
$E(a,x)$ parabolic cylinder function 693
$\mathbf{E}_\nu(z)$ Weber's function 498
$E_\nu^{(m)}(z)$ Weber parabolic cylinder function 509
$E(m)$ complete elliptic integral of the second kind 590
Ei(x) exponential integral 228
$E_1(z)$ exponential integral 228
$E[g(X)]$ expected value operator for the function g(x) 928
Ein(z) modified exponential integral 228
$E_n$ Euler number 804
$E_n(x)$ Euler polynomial 804
$E_n(z)$ exponential integral 228
erf z error function 297
erfc z complementary error function 297
exp z=$e^z$ exponential function 69
exsec A, exsecant A 78
$f_{e,r}$, $f_{o,r}$ joining factors for Mathieu functions 735
$F(a, b; c; z)$ hypergeometric function 556
F($\phi$ \a) elliptic integral of the first kind 589
$F_L(\eta,\rho)$ Coulomb wave function (regular) 538
FPP fundamental period parallelogram 629
${}_n F_m(a_1,...,a_n; b_1,...,b_m; z)$ generalized hyper-geometric function 556
$g_2$, $g_3$ invariants of Weierstrass elliptic functions 629
$g_{e,r}$, $g_{o,r}$ joining factors for Mathieu functions 740
$g(x,y,\rho)$ bivariate normal probability function 936
Gi(z) related Airy function 448
$G_L(\eta,\rho)$ Coulomb wave function (irregular or logarithmic) 538
$G_n(p,q,x)$ Jacobi polynomial 774
gd(z) Gudermannian 77
$h_n^{(1)}(z)$ spherical Bessel function of the third kind 437
hav A haversine A 78
$\mathbf{H}_\nu(z)$ Struve's function 496
Hi(z) related Airy function 448
$He_n(z)$ Hermite polynomial 775
$H_\nu^{(n)}(z)$ Bessel function of the third kind (Hankel) 358
$Hh_n(x)$ Hh (probability) function 300, 691
$H_n(x)$ Hermite polynomial 775
$H(m, n, x)$ confluent hypergeometric function 695
$I_\nu(z)$ modified Bessel function 374
$\sqrt{\pi/2}I_{n+1/2}(z)$ modified spherical Bessel function of the first kind 443
$\sqrt{\pi/2}I_{-n-1/2}(z)$ modified spherical Bessel function of the second kind 443
$I(u, p)$ incomplete gamma function (Pearson's form) 262
$I_x(a,b)$ incomplete beta function 263
$\cal{I} z$ imaginary part of z (=y) 16
$\mathrm{i}^n$ erfc z repeated integral of the error function 299
$j_n(z)$ spherical Bessel function of the first kind 437
$\mathbf{J}_\nu(z)$ Anger's function 498
$J_\nu(z)$ Bessel function of the first kind 358
k modulus of Jacobian elliptic functions 590
k' complementary modulus 590
$k_nu(z)$ Bateman's function 510
$\mathrm{Ki}_\nu(z)$ repeated integrals of $K_0(z)$ 483
$\sqrt{\pi/2}I_{n+1/2}(z)$ modified spherical Bessel function of the third kind 443
$K_\nu(z)$ modified Bessel function 374
$K(m)$ complete elliptic integral of the first kind 590
$\mathrm{ker}_\nu(x)$, $\mathrm{kei}_\nu(x)$ Kelvin functions 379
li(x) logarithmic integral 228
lim limit 13
$\log_{10}x$ common (Briggs) logarithm 68
$\log_a z$ logarithm of z to base a 67
$\ln z (=\log_e z)$ natural, Naperian or hyperbolic logarithm 68
$\cal{L} F(t)=f(s)$ Laplace transform 1020
$L(h,k,\rho)$ cumulative bivariate normal probability function 936
$L_n(x)$ Laguerre polynomial 775
$L_n^{(\alpha)}(x)$ generalized Laguerre polynomial 775
$\mathbf{L}_\nu(z)$ modified Struve function 498
$m = \mu_1^\prime$ mean 928
m parameter (elliptic functions) 569
$m_1$ complementary parameter 569
$M(a,b,z)$ Kummer's confluent hypergeometric function 504
$Mc_r^{(j)}(z, q)$ modified Mathieu function 733
$Ms_r^{(j)}(z, q)$ modified Mathieu function 733
$M_{g,\mu}(z)$ Whittaker function 505
n characteristic of the elliptic integral of the third kind 590
nc, nd, ns, Jacobian elliptic functions 570
$O(v_n)=u_n$, $u_n$ is of the order of $v_n$ ($u_n/v_n$ is bounded) 15
$o(v_n)=u_n$, $\lim_{n\to\infty} u_n/v_n=0$ 259
$O_n(z)$ Neumann's polynomial 363
$p(n)$ number of partitions 825
$\cal{P}(z)$ Weierstrass elliptic function 629
ph z phase of the complex number z 16
$P(a,x)$ incomplete gamma function 260
$P(\chi^2\diagdown \nu)$ probability of the $\chi^2$-distribution 262, 940
$P_\nu^\mu(z)$ associated Legendre function of the first kind 332
$P(x)$ normal probability function 931
$P_n(z)$ Legendre function (spherical polynomials) 333, 774
$P_n^{*}(y)$ shifted Legendre polynomial 774
$P_n^{(\alpha,\beta)}(x)$ Jacobi polynomial 774
$Pr{X\leq x}$ probability of the event $X\leq x$ 927
q nome 591
$Q(x)=1-P(x)$ normal probability function (tail area) 931
$q(n)$ number of partitions into distinct integer summands 825
$Q_\nu^\mu(z)$ associated Legendre function of the second kind 332
$P_n(z)$ Legendre function of the second kind 334
$\cal{R} z$ real part of z (=x) 16
$R_{mn}^{(\nu)}(c,\xi)$ radial spheroidal wave function 753
$S_n^{(m)}$ Stirling number of the first kind 824
$\cal{S}_n^{(m)}$ Stirling number of the second kind 824
$se_r(z,q)$ Mathieu function 725
sc, sd, sn, Jacobian elliptic functions 569
$S(z)$ Fresnel integral 300
$S_1(z)$, $S_2(z)$ Fresnel integrals 300
$Se_r(z,q)$ modified Mathieu function 733
$S(x,a)$ generalized Fresnel integral 262
Shi(z) hyperbolic sine integral 231
Si(z) sine integral 231
$S_n(x)$ Chebyshev polynomial of the first kind 774
Sih(z) hyperbolic sine integral 231
$S_{mn}^{(\nu)}(c,\eta)$ angular spheroidal wave function 753
si(z) sine integral 232
sin z, cos z, tan z circular functions 71
- cot z, sec z, csc z 72
sinh z, cosh z, tanh z hyperbolic functions 83
- coth z, sech z, csch z 83
$T(m,n,r)$ Toronto function 509
$T_n(x)$ Chebyshev polynomial of the first kind 774
$T_n^{*}(x)$ shifted Chebyshev polynomial of the first kind 774
$U(a, b, z)$ Kummer's confluent hypergeometric function 504
$U_n(x)$ Chebyshev polynomial of the second kind 774
$U_n^{*}(x)$ shifted Chebyshev polynomial of the second kind 774
$U(a,x)$ Weber parabolic cylinder function 687
vers A, versine A 78
$Y(a,x)$ Weber parabolic cylinder function 687
$w(z)$ error function 297
$W(a,x)$ Weber parabolic cylinder function 692
$W_{\kappa,\mu}(z)$ Whittaker function 505
$W\{f(x), g(x)\}(=f(x)g'(x)-f'(x)g(x))$ Wronskian relation 505
$[x_0,x_1,\dots,x_k]$ divided difference 877
$y_n(z)$ spherical Bessel function of the second kind 437
$Y_\nu(z)$ Bessel function of the second kind 358
$Y_n^m(\theta,\phi)$ surface harmonic of the first kind 332
$Z(x)$ normal probability density function 931

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jonsson.eu

M. Abramowitz and I. A. Stegun.

Handbook of mathematical functions.

Subject Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

Z

Index of Notations