Index A

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

Abelian functions §21.8
Abel means ¶ ‣ §1.15(iii)
Abel–Plana formula §2.10(i)
Abel summability ¶ ‣ §1.15(i), ¶ ‣ §1.15(iv)
absolute error §3.1(v)
acceleration of convergence
- definition §3.9(i)
- for sequences §3.9, §3.9(vi)
- for series §3.9, §3.9(vi)
- limit-preserving §3.9(i)
accumulation point ¶ ‣ §1.9(ii)
acoustics
- canonical integrals §36.14(iv)
additive number theory Ch.27, §27.14(vi)
- Dedekind modular function §27.14(iv)
- Dedekind sum §27.14(iii)
- discriminant function §27.14(vi)
- Euler’s pentagonal number theorem §27.14(ii)
- Goldbach conjecture §27.13(ii)
- Jacobi’s identities §27.13(iv)
- notation §27.1
- partition function §27.13(i)
  - unrestricted §27.14(i)
- Ramanujan’s identity §27.14(v)
- Ramanujan’s tau function §27.14(vi)
- representation by squares §27.13(iv)
- Waring’s problem §27.13(iii)
aerodynamics
- Struve functions §11.12
affine Weyl groups
- Painlevé equations §32.7(viii)
Airy functions §9.1
- analytic properties §9.2(i)
- applications
  - mathematical §9.15
  - physical §9.16, §9.16
  - ship waves §36.13
- approximations
  - expansions in Chebyshev series §9.19(ii)
  - in terms of elementary functions §9.19(i)
  - in the complex plane §9.19(iii)
- asymptotic expansions §9.7, §9.7(v)
  - error bounds §9.7(iii), §9.7(iv)
  - exponentially-improved §9.7(v)
- computation §9.17, §9.17(v)
- connection formulas §9.2(v)
- definitions §9.2(i)
- differential equation §9.2(i)
  - for products §9.11(i)
  - initial values §9.2(ii)
  - numerically satisfactory solutions §9.2(iii)
  - Riccati form §9.2(vi)
- Dirac delta ¶ ‣ §1.17(ii)
- envelope functions §2.8(iii)
- generalized, see generalized Airy functions.
- graphics §9.3, §9.3(ii)
- incomplete §9.14
- integral identities §36.9
- integral representations §9.11(iii), §9.5(i), §9.5(ii)
- integrals
  - approximations §9.19(i), §9.19(ii), §9.19(ii), §9.19(i), §9.19(ii), §9.19(iii)
  - asymptotic approximations §9.10(ii)
  - definite §9.10(iv)
  - indefinite §9.10(i)
  - of products §9.11(iv), §9.11(v)
  - repeated §9.10(viii)
  - tables §9.18(v)
- Laplace transforms §9.10(v)
- Maclaurin series §9.4
- Mellin transform §9.10(vi)
- modulus and phase
  - asymptotic expansions §9.8(iv), §9.8(iv)
  - definitions §9.8(i)
  - graphs §9.3(i)
  - identities §9.8(ii)
  - monotonicity §9.8(iii)
  - relation to Bessel functions §9.8(i)
  - relation to zeros §9.9(ii)
- notation §9.1
- products
  - differential equation §9.11(i)
  - integral representations §9.11(iii)
  - integrals §9.11(iv), §9.11(v)
  - Wronskian §9.11(ii)
- relations to other functions
  - Bessel functions §9.6(i), §9.6(ii)
  - confluent hypergeometric functions §13.18(iii), §13.6(iii), §9.6(iii)
  - Hankel functions §9.6(i), §9.6(ii)
  - modified Bessel functions §9.6(i), §9.6(ii)
- relation to umbilics §36.2(ii)
- Stieltjes transforms §9.10(vii)
- tables
  - complex variables §9.18(iii)
  - integrals §9.18(v)
  - real variables §9.18(ii)
  - zeros §9.18(iv), §9.9(v)
- Wronskians §9.2(iv)
- zeros
  - asymptotic expansions §9.9(iv)
  - computation ¶ ‣ §3.3(v), §9.17(v)
  - differentiation §9.9(iii)
  - relation to modulus and phase §9.9(ii)
  - tables §9.18(iv), §9.9(v)
Airy’s equation, see Airy functions, differential equation.
Airy transform §9.10(ix)
Aitken’s $\Delta^{2}$ -process
- for sequences §3.9(iii)
- iterated §3.9(iii)
algebraic curves
- Riemann surface §21.10(ii), §21.7(i), §21.7(iii)
algebraic equations
- parametrization via Jacobian elliptic functions §22.18(i)
  - spherical trigonometry §22.18(iii)
  - uniformization §22.18(iii)
algebraic Lamé functions §29.17(ii)
Al-Salam–Chihara polynomials §18.28(iv)
alternant
- determinant §1.3(ii)
amplitude ( $\mathop{\mathrm{am}\/}\nolimits$ ) function §22.16(i)
- applications §22.19(i)
- approximations
  - small $k,k^{{\prime}}$ ¶ ‣ §22.16(i)
  - small $x$ ¶ ‣ §22.16(i)
- computation §22.20(vi)
- definition ¶ ‣ §22.16(i)
- Fourier series ¶ ‣ §22.16(i)
- integral representation ¶ ‣ §22.16(i)
- quasi-periodicity ¶ ‣ §22.16(i)
- relation to elliptic integrals ¶ ‣ §22.16(i)
- relation to Gudermannian function ¶ ‣ §22.16(i)
- special values ¶ ‣ §22.16(i)
- tables §22.21
analytic continuation §1.10(ii)
- by reflection ¶ ‣ §1.10(ii)
analytic function ¶ ‣ §1.9(ii)
- at infinity §1.9(iv)
- in a domain ¶ ‣ §1.9(ii)
- singularities §1.10(iii)
- zeros ¶ ‣ §1.10(i)
Anger function, see Anger–Weber functions.
Anger–Weber functions §11.10
- analytic properties §11.10(i)
- asymptotic expansions
  - large argument §11.11(i), §11.11(i)
  - large order §11.11(ii)
- computation §11.13(i), ¶ ‣ §3.6(vi)
- definitions §11.10(i)
- derivatives §11.10(ix)
- differential equation §11.10(ii)
- graphics Figure 11.10.1, Figure 11.10.1, Figure 11.10.2, Figure 11.10.2, Figure 11.10.3, Figure 11.10.3, Figure 11.10.4, Figure 11.10.4
- incomplete §11.14(v)
- integral representations §11.10(i)
- integrals §11.10(x)
- interrelations §11.10(v)
- Maclaurin series §11.10(iii)
- notation §11.1
- order §11.1
- recurrence relations §11.10(ix)
- relations to other functions
  - Fresnel integrals §11.10(vi)
  - Lommel functions §11.10(vi), §11.10(vi)
  - Struve functions §11.10(vi)
- series expansions
  - power series §11.10(iii)
  - products of Bessel functions §11.10(viii)
- special values §11.10(vii)
- sums §11.10(x)
- tables §11.14(iv)
angle between arcs ¶ ‣ §1.9(iv)
angular momenta §34.2
angular momentum
- generalized hypergeometric functions §16.24(iii)
angular momentum coupling coefficients, see $3j$ symbols, $6j$ symbols, and $9j$ symbols.
angular momentum operator
- spherical coordinates §14.30(iv)
annulus §1.10(iii)
antenna research
- Lamé functions §29.19(i)
Appell functions §16.13
- analytic continuation §16.15
- applications
  - physical §16.24
- computation Ch.16
- definition §16.13, §16.13
- integral representations §16.15
- integrals §16.15
  - inverse Laplace transform §16.15
- notation §16.13
- partial differential equations §16.14(i)
- relations to hypergeometric functions §16.16(i)
- relation to Legendre’s elliptic integrals §19.5
- relation to symmetric elliptic integrals §19.25(vii)
- transformations of variables §16.16, §16.16(ii)
  - quadratic §16.16(ii)
  - reduction formulas §16.16(i)
approximation techniques
- Chebyshev-series expansions ¶ ‣ §3.11(ii)
- least squares ¶ ‣ §3.11(v), §3.11(v)
- minimax polynomials §3.11(i)
- minimax rational functions §3.11(iii)
- Padé ¶ ‣ §3.11(iv), §3.11(iv)
- splines §3.11(vi)
arc length
- Jacobian elliptic functions §22.18(i)
arc(s) §1.9(iii)
- angle between ¶ ‣ §1.9(iv)
area of triangle ¶ ‣ §10.22(iv)
argument principle, see phase principle.
arithmetic Fourier transform §27.17
arithmetic-geometric mean §19.8(i)
- hypergeometric function §15.17(iv)
- integral representations §19.8(i)
- Jacobian elliptic functions §22.20(ii)
- Legendre’s elliptic integrals §19.8(i), §19.8(i)
- symmetric elliptic integrals §19.22(ii)

arithmetic mean §1.2(iv), §1.7(iii)
arithmetic progression ¶ ‣ §1.2(ii)
arithmetics
- complex §3.1(v)
- exact rational §3.1(iii)
- floating-point §3.1(i)
- interval §3.1(ii)
- level-index §3.1(iv)
Askey–Gasper inequality
- Jacobi polynomials ¶ ‣ §18.38(ii)
Askey polynomials ¶ ‣ §18.33(iv)
Askey scheme for orthogonal polynomials Figure 18.21.1, Figure 18.21.1
Askey–Wilson class orthogonal polynomials §18.28, §18.28(viii)
- as eigenfunctions of a $q$ -difference operator §18.28(i)
- asymptotic approximations §18.29
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1
- orthogonality properties §18.28(i)
- representation as $q$ -hypergeometric functions §18.28(i), §18.28(viii)
Askey–Wilson polynomials §18.28(ii)
- asymptotic approximations §18.29
- relation to $q$ -hypergeometric functions §18.28(ii), §18.28(viii)
associated Anger–Weber function, see Anger–Weber functions.
associated Laguerre functions §33.22(v)
associated Legendre equation §14.2(ii), §14.21(i)
- exponent pairs §14.2(iii)
- numerically satisfactory solutions §14.2(iii), §14.21(ii)
- singularities §14.2(iii)
- standard solutions §14.2(ii), §14.21(i), ¶ ‣ §14.3(ii)
associated Legendre functions §14.1, see also Ferrers functions.
- addition theorems §14.18(ii), §14.28(i)
- analytic continuation §14.24
- analytic properties §14.21(i)
- applications Ch.14, §14.31(iii)
- asymptotic approximations, see uniform asymptotic approximations.
- behavior at singularities §14.21(iii), §14.8(i)
- computation §14.32
- connection formulas §14.21(iii), §14.9(iii)
- continued fractions §14.14
- cross-products §14.2(iv)
- definitions §14.21(i), §14.3(ii), §14.3(iii)
- degree §14.1
- derivatives §14.10
  - with respect to degree or order §14.11
- differential equation, see associated Legendre equation.
- expansions in series of §14.18(i)
- generalized §14.29
- generating functions §14.21(iii), §14.7(iv)
- graphics §14.22, §14.4(iii), §14.4(iv)
- Heine’s formula §14.28(ii)
- hypergeometric representations §14.21(iii), §14.3, §14.3(ii), §14.3(iii)
- integer degree and order §14.21(iii), §14.7, §14.7(iv)
- integer order §14.21(i), §14.6, §14.6(ii)
- integral representations §14.12(ii), §14.25
- integrals
  - definite §14.17(ii), §14.17(iii), §14.17(iv)
  - Laplace transforms §14.17(v)
  - Mellin transforms §14.17(vi)
  - products §14.17(iv)
- notation §14.1
- of the first kind ¶ ‣ §14.3(ii)
- of the second kind ¶ ‣ §14.3(ii)
- Olver’s §14.21(i), ¶ ‣ §14.3(ii)
- order §14.1
- orthogonality §14.17(iii)
- principal values (or branches) §14.21(i)
- recurrence relations §14.10, §14.21(iii)
- relations to other functions
  - elliptic integrals §14.5(v)
  - Gegenbauer function §14.3(iv)
  - hypergeometric function §14.3(ii), §14.3(iii), §15.9(iv)
  - Jacobi function §14.3(iv)
  - Legendre polynomials §14.7(i)
- Rodrigues-type formulas §14.7(ii)
- special values §14.5(iii), §14.5(v)
- sums §14.18, §14.18(iv), §14.28
- tables §14.33
- uniform asymptotic approximations
  - large degree §14.15(iii), §14.15(v), §14.26
  - large order §14.15(i), §14.15(ii), §14.26
- values on the cut §14.23
- Whipple’s formula §14.9(iv)
- Wronskians §14.2(iv), §14.21(iii)
- zeros §14.16(iii), §14.27
associated orthogonal polynomials §18.30, ¶ ‣ §18.30
- corecursive §18.30
- Jacobi ¶ ‣ §18.30
- Legendre ¶ ‣ §18.30
astrophysics
- error functions and Voigt functions §7.21
- Heun functions and Heun’s equation §31.17(ii)
asymptotic and order symbols §2.1(i)
- definition §2.1(i)
- differentiation §2.1(ii)
- integration §2.1(ii)
asymptotic approximations and expansions, see also asymptotic approximations of integrals, asymptotic approximations of sums and sequences, asymptotic solutions of difference equations, asymptotic solutions of differential equations, and asymptotic solutions of transcendental equations.
- algebraic operations §2.1(iii)
- cases of failure §2.11(i), §2.11(i), §2.6(i)
- differentiation §2.1(iii)
- double asymptotic properties
  - Bessel functions §10.41(v)
  - Hankel functions §10.41(v)
  - Kelvin functions §10.69
  - modified Bessel functions §10.41(iv)
  - parabolic cylinder functions §12.10(vi)
- exponentially-improved expansions §2.11(iii), §2.11(v)
- generalized §2.1(v)
- hyperasymptotic expansions §2.11(v)
- improved accuracy via numerical transformations §2.11(vi)
- integration §2.1(iii)
- logarithms of §2.1(iii)
- null §2.1(iii)
- numerical use of §2.11(i), §2.11(vi)
- Poincaré type §2.1(iii)
- powers of §2.1(iii)
- re-expansion of remainder terms §2.11(iii), §2.11(vi)
- reversion of ¶ ‣ §2.2
- Stokes phenomenon §2.11(iv)
- substitution of §2.1(iii)
- uniform §2.1(iv)
- uniqueness §2.1(iii)
- via connection formulas §2.11(ii)
asymptotic approximations of integrals ¶ ‣ §2.2, §2.6(iv)
- Bleistein’s method §2.3(v)
- Chester–Friedman–Ursell method §2.4(v)
- coalescing critical points §2.4(v), §2.4(vi)
- coalescing peak and endpoint §2.3(v)
- coalescing saddle points §2.4(v)
- distributional methods §2.6, §2.6(iv)
- Fourier integrals §2.3(i)
- Haar’s method §2.4(ii)
- integration by parts §2.3(i)
- inverse Laplace transforms §2.4(i), §2.4(ii)
- Laplace’s method §2.3(iii), §2.4(iii)
- Laplace transforms §2.3(i)
- Mellin transform methods §2.5
  - extensions §2.5(ii)
- method of stationary phase §2.3(iv)
  - extensions §2.3(iv)
- method of steepest descents §2.4(iv)
- multidimensional integrals §2.5(ii)
- Stieltjes transforms §2.6(ii)
  - generalized §2.6(ii)
- Watson’s lemma §2.3(ii), §2.4(i)
  - generalized §2.3(ii)
asymptotic approximations of sums and sequences §2.10, ¶ ‣ §2.10(iv)
- Abel–Plana formula §2.10(i)
- Darboux’s method ¶ ‣ §2.10(iv), §2.10(iv)
- entire functions §2.10(iii)
- Euler–Maclaurin formula ¶ ‣ §2.10(i), §2.10(i)
- summation by parts §2.10(ii)
asymptotic scale or sequence §2.1(v)
asymptotic solutions of difference equations §2.9, §2.9(iii)
- characteristic equation §2.9(i)
- coincident characteristic values §2.9(ii)
- Liouville–Green (or WKBJ) type approximations §2.9(iii)
- transition points §2.9(iii)
- turning points §2.9(iii)
- with a parameter §2.9(ii), §2.9(iii)
asymptotic solutions of differential equations §2.6(iv), §2.8(vi)
- characteristic equation §2.7(ii)
- coincident characteristic values §2.7(ii)
- error-control function ¶ ‣ §2.7(iii)
- Fabry’s transformation §2.7(ii)
- irregular singularities of rank 1 §2.7(ii)
- Liouville–Green approximation theorem §2.7(iii)
- Liouville–Green (or WKBJ) approximations ¶ ‣ §2.7(iii), §2.7(iii)
- numerically satisfactory solutions §2.7(iv)
- resurgence §2.11(v), §2.7(ii)
- with a parameter §2.8, §2.8(vi)
  - classification of cases §2.8(i)
  - coalescing transition points §2.8(vi)
  - connection formulas across transition points §2.8(v)
  - in terms of Airy functions §2.8(iii)
  - in terms of Bessel functions of fixed order §2.8(iv)
  - in terms of Bessel functions of variable order §2.8(vi)
  - in terms of elementary functions §2.8(ii)
  - Liouville transformation §2.8(i)
  - transition points §2.8(i)
  - turning points §2.8(i)
asymptotic solutions of transcendental equations §2.2
- Lagrange’s formula ¶ ‣ §2.2
atomic photo-ionization
- Coulomb functions ¶ ‣ §33.22(ii)
atomic physics
- Coulomb functions ¶ ‣ §33.22(ii)
- error functions §7.21
atomic spectra
- Coulomb functions ¶ ‣ §33.22(ii)
atomic spectroscopy
- $3j,6j,9j$ symbols §34.12
attractive potentials
- Coulomb functions ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii), ¶ ‣ §33.22(ii)
auxiliary functions for Fresnel integrals
- approximations §7.24(i)
- asymptotic expansions §7.12(ii)
- computation §7.22(i)
- definitions §7.2(iv)
- derivatives §7.10
- integral representations §7.7(ii)
  - Mellin–Barnes integrals ¶ ‣ §7.7(ii)
- symmetry §7.4
auxiliary functions for sine and cosine integrals
- analytic continuation §6.4
- approximations §6.20(i)
- asymptotic expansions §6.12(ii)
  - exponentially-improved §6.12(ii)
- Chebyshev-series expansions §6.20(ii)
- computation §6.18(ii)
- definition §6.2(iii)
- integral representations §6.7(iii)
- principal values §6.4
- relation to confluent hypergeometric functions ¶ ‣ §6.11
- tables §6.19(ii)
axially symmetric potential theory §19.18(ii)