Index
- Abel means §1.15(iii)
- Abel summability §1.15(i), §1.15(iv)
- Abel–Plana formula §2.10(i)
- Abelian functions §21.8
- absolute error §3.1(v)
- Absolutely continuous integration measure §1.4(v)
- acceleration of convergence
- 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
-
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)
- computation §9.17—§9.17(v)
- connection formulas §9.2(v)
- definitions §9.2(i)
- differential equation §9.2(i)
- 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
- Laplace transforms §9.10(v)
- Maclaurin series §9.4
- Mellin transform §9.10(vi)
- modulus and phase
- 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)
- relation to umbilics §36.2(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)
- Stieltjes transforms §9.10(vii)
- tables
- Wronskians §9.2(iv)
- zeros
- Airy transform §9.10(ix)
- Airy’s equation, see Airy functions, differential equation.
-
Aitken’s
-process
- Al-Salam–Chihara polynomials §18.28(iv)
-
algebraic curves
- Riemann surface 2nd item, §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)
-
parametrization via Jacobian elliptic functions §22.18(i)
- algebraic Lamé functions §29.17(ii)
-
almost Mathiew equation
- singular continuous spectra §1.18(vii)
-
alternant
- determinant §1.3(ii)
-
amplitude (
) function §22.16(i)
-
analytic continuation §1.10(ii)
- by reflection §1.10(ii)
-
analytic continuation of matrix elements of the resolvent onto higher Riemann sheets
- dilatation transformations §1.18(viii)
-
analytic continuation onto higher Riemann sheets
- matrix elements of the resolvent §1.18(viii)
-
analytic function §1.9(ii)
- at infinity §1.9(iv)
- in a domain §1.9(ii)
- singularities §1.10(iii)
- zeros §1.10(i)
-
Anderson localization
- random potentials §1.18(vii)
- 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 11.10.1, Figure 11.10.2, Figure 11.10.2, Figure 11.10.2, Figure 11.10.3, Figure 11.10.3, Figure 11.10.3, Figure 11.10.4, 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
symbols, 
symbols, and 
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)
- relation to Legendre’s elliptic integrals §19.5
- relation to symmetric elliptic integrals §19.25(vii)
- relations to hypergeometric functions §16.16(i)
-
transformations of variables §16.16—§16.16(ii)
- quadratic §16.16(ii)
- reduction formulas §16.16(i)
- approximation techniques
-
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 mean §1.2(iv), §1.7(iii)
- arithmetic progression §1.2(ii)
-
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)
- arithmetics
- Askey polynomials §18.33(iv)
- Askey scheme for orthogonal polynomials Figure 18.21.1, Figure 18.21.1, Figure 18.21.1
- Askey–Gasper inequality §18.14(iv), §18.38(ii)
-
Askey–Wilson class orthogonal polynomials §18.28—§18.28(xi)
-
as eigenfunctions of a
-difference operator §18.28(i)
- asymptotic approximations §18.29
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1, Figure 18.21.1
- limit relations §18.28(x)
- orthogonality properties §18.28(i)
-
representation as -hypergeometric functions §18.28(i)—§18.28(xi)
-
as eigenfunctions of a
-
Askey–Wilson polynomials §18.28(ii), see also Askey–Wilson class orthogonal
polynomials.
- and Zhedanov algebra §18.38(iii)
- asymptotic approximations §18.29
- duality §18.28(ii)
- nonsymmetric §18.38(iii)
-
-difference equation §18.28(ii)
- recurrence relation §18.28(ii)
-
relation to -hypergeometric functions §18.28(ii)—§18.28(xi)
- associated Anger–Weber function, see Anger–Weber functions.
- associated Hermite polynomials §18.30(iv)
- associated Laguerre functions §33.22(v)
- associated Laguerre polynomials §18.30(iii)
-
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(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
- 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)—§14.2(iv), §14.21(iii)
- zeros §14.16(iii), §14.27
-
associated orthogonal polynomials §18.2(x), §18.30—§18.30
- and corecursive OP’s §18.2(x), §18.30(vi)
- Hermite §18.30(iv)
- Jacobi §18.30(i)
- Laguerre §18.30(iii)
- Legendre §18.30(ii)
- Meixner–Pollaczek §18.30(v)
- monic §18.2(x), §18.30(vii)
- type 2 Pollaczek §18.30(viii)
- ultraspherical §18.30(viii)
-
astrophysics
- error functions and Voigt functions §7.21
- Heun functions and Heun’s equation §31.17(ii)
- asymptotic and order symbols §2.1(i)
-
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)—§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 transforms §2.3(i)
- Laplace’s method §2.3(iii)—§2.4(iii)
- Mellin transform §2.3(vi)
- Mellin transform methods §2.5
-
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)—§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)
- asymptotic scale or sequence §2.1(v)
- asymptotic solutions of difference equations §2.9—§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 (or WKBJ) approximations §2.7(iii)—§2.7(iii)
- Liouville–Green approximation theorem §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)—§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
-
symbols §34.12
-
-
attractive potentials
- Coulomb functions §33.22(ii), §33.22(ii), §33.22(ii)
- auxiliary functions for Fresnel integrals
-
auxiliary functions for sine and cosine integrals
- analytic continuation §6.4
- approximations 4th item
-
asymptotic expansions §6.12(ii)
- exponentially-improved §6.12(ii)
- Chebyshev-series expansions 5th item
- 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)
