Subject Index
Note: pages omitted from this
electronic edition are not hyperlinked
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
- 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 ..... 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
- Arithmetic-geometric mean ..... 571, 577, 578, 580, 598, 601
- Arithmetic mean ..... 10
- Arithmetic progression ..... 10
- Associated Legendre functions ..... 331
- Asymptotic expansions ..... 15
B
- Basic numbers ..... 822
- Bateman function ..... 510
- Bernoulli numbers ..... 804
- 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
- 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
- Cartesian form ..... 16
- Catalan's constant ..... 807
- Cauchy-Riemann equation ..... 17
- Cauchy's inequality ..... 11
- Characteristic function ..... 928
- Chebyshev integration ..... 887
- Chebyshev polynomials ..... 486, 561, 774
- Chebyshev's inequality ..... 11
- Chi-square distribution function
- 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
- 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
- Digamma function ..... 258
- 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
- Double precision operations ..... 21
E
- Economization of series ..... 791
- Edgeworth asymptotic expansion ..... 935, 955
- Einstein functions ..... 999
- Elliptic functions
- Jacobian ..... 567
- Weierstrass ..... 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
- Euler-Maclaurin formulas ..... 806
- Euler-Maclaurin summation formula ..... 16, 22, 806, 886
- Euler numbers ..... 804
- 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
- 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
- Factorization
- 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
- 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
- Generalized hypergeometric function ..... 362, 377, 556
- Generalized Laguerre polynomials ..... 771
- Generalized mean ..... 10
- Geometric mean ..... 10
- Geometric progression ..... 10
- Gill's method ..... 896
- Gudermannian ..... 77
H
- Hankel functions ..... 358, 379, 510
- Hankel's contour integral ..... 255
- Harmonic analysis ..... 202, 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
- Heuman's lambda function ..... 595
- graph of ..... 595
- table of ..... 622
- Hh function ..... 300, 691
- Hölder's inequality for integrals ..... 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
- 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
- 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
- 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
- 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
- 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
- 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
- Lagrange differentiation formula ..... 882
- Lagrange integration coefficients ..... 886
- Lagrange interpolation coefficients ..... 878
- Lagrange interpolation formula ..... 878
- Lagrange's expansion ..... 14
- Laguerre integration ..... 890
- abscissas and weight factors for ..... 923
- Laguerre polynomials ..... 509, 510, 773
- 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
- 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
- Legendre's differential equation ..... 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
M
- 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
- characteristic values ..... 722, 748
- asymptotic expansions of ..... 726
- determination of ..... 722
- 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
- Modified spherical Bessel functions ..... 443, 453, 498
- addition theorems for ..... 445
- ascending series for ..... 443
- computation of ..... 453
- 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
N
- 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
- Normal probability integral
- repeated integrals of ..... 934
- Number theoretic functions ..... 826
O
- 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
- 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
- Partitions ..... 825
- into distinct parts ..... 825
- tables of ..... 831, 836
- unrestricted ..... 825
- Pearson's form of the incomplete gamma function ..... 262
- Pentagamma function ..... 260
- 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
- Primitive roots ..... 827
- 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
- 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
- Quadratic equation, solution of ..... 17, 19
- Quartic equation, resolution into quadratic factors ..... 17, 20
R
- 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
- 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
- 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
- Shifted Legendre polynomials ..... 774
- 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
- 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
- Taylor expansion ..... 880
- Taylor's formula ..... 14
- Tesseral harmonics ..... 332
- Tetrachoric functions ..... 934
- Tetragamma function ..... 260
- 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
- Trigonometric functions ..... 71
- Truncated exponential function ..... 70, 262
U
- Ultraspherical polynomials ..... 774
- Unit step function ..... 1020
V
- Variance ..... 928
- Variance-ratio distribution function ..... 946
- Vector-addition coefficients ..... 1006
W
- 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
Index of Notations
- $(a)_n = \Gamma(a+n)/\Gamma(a)$ (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