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

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
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A
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B
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C
    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
    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
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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
      table of ..... 914
    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
      table of ..... 840
    Double precision operations ..... 21
Top of Index

E
    Economization of series ..... 791
    Edgeworth asymptotic expansion ..... 935, 955
    Einstein functions ..... 999
    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
Top of Index

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
      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
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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ドル/\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
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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
      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
Top of Index

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
      tables of ..... 680
    Inverse circular functions ..... 22, 79, 92
      addition and subtraction of ..... 80
      Chebyshev approximations ..... 82
      continued fractions for ..... 81
      differentiation formulas ..... 82
      graph of ..... 79
      indefinite integrals ..... 82
      logarithmic representation ..... 80
      negative arguments ..... 80
      polynomial approximations ..... 81
      real and imaginary parts ..... 80
      relation to inverse hyperbolic functions ..... 80
      series expansions for ..... 81
      table of ..... 203
    Inverse hyperbolic functions ..... 86, 93
      addition and subtraction of ..... 87
      continued fractions for ..... 88
      differentiation formulas ..... 88
      graphs of ..... 86
      indefinite integrals ..... 88
      logarithmic representations ..... 87
      negative arguments ..... 87
      relation to inverse circular functions ..... 87
      series expansions for ..... 88
      tables of ..... 221
Top of Index

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
Top of Index

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
Top of Index

L
    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
    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
    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
Top of Index

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
        generation of ..... 727
        graphs of ..... 728
      characteristic values ..... 722, 748
        asymptotic expansions of ..... 726
        determination of ..... 722
        graph of ..... 724
        power series for ..... 724
      Floquet solutions ..... 727
      other solutions ..... 730
      relation to spheroidal wave equation ..... 722
      solutions involving products of Bessel functions ..... 731
      stability regions of ..... 728
    Mathieu's modified equation ..... 722
      radial solutions of ..... 732
    Maxima ..... 14
    Mean ..... 928
      arithmetic ..... 10
      generalized ..... 10
      geometric ..... 10
      harmonic ..... 10
    Milne's method ..... 896
    Minima ..... 14
    Minkowski's inequality
      for integrals ..... 11
      for sums ..... 11
    Miscellaneous functions ..... 997
    Möbius function ..... 826
    Modified Bessel functions, $I_\nu(z),ドル $K_\nu(z)$ ..... 374
      analytic continuation of ..... 376
      ascending series for ..... 375
      asymptotic expansions for large arguments ..... 377
      connection with Legendre functions ..... 377
      derivatives with respect to order ..... 377
      differential equation ..... 374
      formulas for derivatives ..... 376
      generating function and associated series ..... 376
      graphs of ..... 374
      in terms of hypergeometric functions ..... 377
      integral representations of ..... 376
      limiting forms for small arguments ..... 375
      multiplication theorems for ..... 377
      Neumann series for $K_\nu(z)$ ..... 377
      other differential equations ..... 377
      polynomial approximations ..... 378
      recurrence relations ..... 376
      relations between ..... 375
      tables of ..... 416
      uniform asymptotic expansions for large orders ..... 378
      Wronskian relations ..... 375
      zeros of ..... 377
    Modified Mathieu functions ..... 722
      graphs of ..... 734
    Modified spherical Bessel functions ..... 443, 453, 498
      addition theorems for ..... 445
      ascending series for ..... 443
      computation of ..... 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
      table of ..... 831
Top of Index

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
      tables of ..... 966
    Normal probability integral
      repeated integrals of ..... 934
    Number theoretic functions ..... 826
Top of Index

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
Top of Index

P
    Parabolic cylinder functions $U(a,x),ドル $V(a,x)$ ..... 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
    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,ドル $\nu_2$ ..... 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
Top of Index

Q
    Quadratic equation, solution of ..... 17, 19
    Quartic equation, resolution into quadratic factors ..... 17, 20
Top of Index

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
      solutions of ..... 564
    Riemann's P function ..... 564
      transformation formulas ..... 565
    Ring functions ..... 336
    Rodrigues' formula ..... 334, 773, 785
    Roots
      computation of ..... 19, 89
      graph of ..... 19
      of complex numbers ..... 17, 20
      tables of ..... 24, 223
    Runge-Kutta methods ..... 896
Top of Index

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),ドル $h_n^{(2)}(z)$ ..... 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
Top of Index

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
Top of Index

U
    Ultraspherical polynomials ..... 774
    Unit step function ..... 1020
Top of Index

V
Top of Index

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},ドル $\cal{P}^\prime,ドル $\zeta$ ..... 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}|,ドル $|\cal{P}^\prime|,ドル $|\zeta|$ ..... 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
Top of Index

Z
    Zeta function
Top of Index

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,ドル $bei_\nu x,ドル 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),ドル $C_2(z)$ 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,ドル $e_2,ドル $e_3,ドル 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},ドル $f_{o,r}$ 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,ドル $g_3$ invariants of Weierstrass elliptic functions ..... 629
    $g_{e,r},ドル $g_{o,r}$ 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),ドル $\mathrm{kei}_\nu(x)$ 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,ドル $u_n$ is of the order of $v_n$ ($u_n/v_n$ is bounded) ..... 15
    $o(v_n)=u_n,ドル $\lim_{n\to\infty} u_n/v_n=0$ ..... 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),ドル $S_2(z)$ 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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