M. Abramowitz and I. A. Stegun.
Handbook of mathematical functions.

Subject Index

Note: pages omitted from this electronic edition are not hyperlinked

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
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 )
Asymptotic expansions ..... 15

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
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

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

Economization of series ..... 791
Edgeworth asymptotic expansion ..... 935 , 955
Einstein functions ..... 999
Elliptic functions
Jacobian ..... 567
(see Jacobian elliptic functions )
Weierstrass ..... 627
(see Weierstrass elliptic functions )
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-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

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

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
(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

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

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

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

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

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

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

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

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

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

$(a)_n = \Gamma(a+n)/\Gamma(a)$ (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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