Index W

Waring’s problem
- number theory §27.13(iii)
water waves
- Kelvin’s ship-wave pattern §36.13—§36.13
- Riemann theta functions §21.9, §21.9—§21.9
- Struve functions §11.12
Watson integrals
- Appell functions §16.24(i)
- generalized hypergeometric functions §16.24(i)
Watson’s ${{}_{3}F_{2}}$ sum
- Andrews’ terminating $q$ -analog §17.7(iii)
- Gasper–Rahman $q$ -analog §17.7(iii)
Watson’s expansions
- theta functions §20.8
Watson’s identities
- theta functions §20.7(v)
Watson’s lemma
- asymptotic expansions of integrals §2.3(ii), §2.4(i)
Watson’s sum
- generalized hypergeometric functions §16.4(ii)
wave acoustics
- generalized exponential integral §8.24(iii)
wave equation, see also water waves.
- Bessel functions and modified Bessel functions §10.73(i)
- confluent hypergeometric functions §13.28(i)
- ellipsoidal coordinates §29.18(ii)
- Mathieu functions 1st item, §28.33(ii)
- oblate spheroidal coordinates §30.14(i)—§30.14(v)
- paraboloidal coordinates §13.28(i)
- prolate spheroidal coordinates §30.13—§30.13(v)
- separation constants §29.18(i)
- spherical Bessel functions §10.73(ii)
- sphero-conal coordinates §29.18(i)
- symmetric elliptic integrals §19.18(ii)
wave functions
- paraboloidal §28.31(iii)
- potential §15.19(i)
waveguides §10.73(i)
Weber function, see Anger–Weber functions.
Weber parabolic cylinder functions, see parabolic cylinder functions.
Weber transform
- Bessel functions §10.22(v)
Weber–Schafheitlin discontinuous integrals
- Bessel functions §10.22(iv)
Weber’s function, see Bessel functions of the second kind.
Weierstrass elliptic functions §23.2(ii)
- addition theorems §23.10(i)
- analytic properties §23.2(ii)
- applications
  - mathematical §23.20
  - physical §23.21—§23.21(iv)
- asymptotic approximations §23.12
- computation §23.22
- definitions §23.2(ii)
- derivatives §23.3(ii)
- differential equations §23.3(ii)
- discriminant §23.3(i)
- duplication formulas §23.10(ii)
- equianharmonic case §23.4(i)—§23.4(i), §23.5(v)
- Fourier series §23.8(i)
- graphics
  - complex variables §23.4(ii)—§23.4(ii)
  - real variables §23.4(i)—§23.4(i)
- homogeneity §23.10(iv)
- infinite products §23.8(iii)
- integral representations §23.11
- integrals §23.14
- lattice §23.2(i)
  - computation §23.22(ii)
  - equianharmonic §23.5(v)
  - generators §23.2(i)
  - invariants §23.3(i)
  - lemniscatic §23.4(i)—§23.4(i), §23.5(iii)
  - notation §23.1
  - points §23.2(i)
  - pseudo-lemniscatic §23.5(iv)
  - rectangular §23.5(ii)
  - rhombic §23.5(iv)—§23.5(iv)
  - roots §23.3(i)
- Laurent series §23.9
- lemniscatic case §23.4(i)—§23.4(i), §23.5(iii)
- $n$ -tuple formulas §23.10(iii)
- notation §23.1
- periodicity §23.2(iii)
- poles §23.2(ii)
- power series §23.9
- principal value §23.8(ii)
- pseudo-lemniscatic case §23.5(iv)
- quarter periods §23.7
- quasi-periodicity §23.2(iii)
- relations to other functions
  - elliptic integrals §23.6(iv)—§23.6(iv)
  - general elliptic functions §23.6(iii)
  - Jacobian elliptic functions §23.6(ii)
  - symmetric elliptic integrals §19.25(vi)
  - theta functions §23.6(i)
- rhombic case §23.5(iv)—§23.5(iv)
- series of cosecants or cotangents §23.8(ii)
- tables §23.23
- zeros §23.13, §23.2(ii)
Weierstrass $M$ -test, see $M$ -test for uniform convergence.
Weierstrass $\wp$ -function, see Weierstrass elliptic functions.
Weierstrass product §1.10(ix)
Weierstrass sigma function, see Weierstrass elliptic functions.
Weierstrass zeta function, see Weierstrass elliptic functions.
weight functions
- cubature §3.5(x)—§3.5(x)
- definition §18.2(i), §3.5(iv)
- Freud §18.32
- least squares approximations §3.11(v)
- logarithmic §3.5(v)—§3.5(v)
- minimax rational approximations §3.11(iii)
- quadrature §3.5(iv)—§3.5(v)
weighted means §1.2(iv)
Weniger’s transformation
- for sequences §3.9(v)

Whipple’s ${{}_{3}F_{2}}$ sum
- Gasper–Rahman $q$ -analog §17.7(iii)
Whipple’s formula
- associated Legendre functions §14.9(iv)
- toroidal functions §14.19(v)
Whipple’s sum
- generalized hypergeometric functions §16.4(ii)
Whipple’s theorem
- Watson’s $q$ -analog §17.9(iii)
Whipple’s transformation
- generalized hypergeometric functions §16.4(iii)
Whittaker functions Ch.13, see also confluent hypergeometric functions.
- addition theorems §13.26(i), §13.26(ii)
- analytic continuation §13.14(ii)
- analytical properties §13.14(i)
- applications
  - Coulomb functions §13.28(ii)
  - groups of triangular matrices §13.27—§13.27
  - physical §13.28(ii), §33.22(v)
  - uniform asymptotic solutions of differential equations §13.27
- asymptotic approximations for large parameters
  - imaginary $\kappa$ and/or $\mu$ §13.20(iv)
  - large $\kappa$ §13.21—§13.21(iv)
  - large $\mu$ §13.20(i)—§13.20(v)
  - uniform §13.20(i)—§13.21(iv)
- asymptotic expansions for large argument §13.19—§13.19
  - error bounds §13.19
  - exponentially-improved §13.19
- computation Ch.13
- connection formulas §13.14(vii)
- continued fractions §13.17
- definitions §13.14(i)
- derivatives §13.15(ii)
- differential equation, see Whittaker’s equation.
- expansions in series of §13.24(i)
- integral representations
  - along the real line §13.16(i)—§13.16(i)
  - contour integrals §13.16(ii)
  - Mellin–Barnes type §13.16(iii)
- integral transforms in terms of §13.23(iv)
- integrals §13.16(i)
  - compendia §13.23(v)
  - Fourier transforms §13.23(ii)
  - Hankel transforms §13.23(iii)—§13.23(iii)
  - Laplace transforms §13.23(i)
  - Mellin transforms §13.23(i)
- interrelations §13.14(vii)
- large argument §2.11(vi)
- limiting forms
  - as $z\rightarrow\infty$ §13.14(iv)
  - as $z\rightarrow 0$ §13.14(iii)
- Morse oscillator §18.39(i)
- multiplication theorems §13.26(iii)
- notation §13.1
- power series §13.14(i)—§13.14(i)
- principal branches (or values) §13.14(i)
- products §13.25
- recurrence relations §13.15(i)
- relations to other functions
  - Airy functions §13.18(iii)
  - Coulomb functions §33.14(ii), §33.14(iii), §33.16(iii), §33.16(v), §33.2(iii)
  - elementary functions §13.18(i)
  - error functions §13.18(ii)
  - incomplete gamma functions §13.18(ii)
  - Kummer functions §13.14(i)
  - modified Bessel functions §13.18(iii)
  - orthogonal polynomials §13.18(v)
  - parabolic cylinder functions §13.18(iv)
- series expansions §13.24—§13.26(ii)
  - addition theorems §13.26(ii)
  - in Bessel functions or modified Bessel functions §13.24(ii)
  - multiplication theorems §13.26(iii)
  - power §13.14(i)—§13.14(i)
- Wronskians §13.14(vi)—§13.14(vi)
- zeros
  - asymptotic approximations §13.22
  - distribution §13.22
  - inequalities §13.22
  - number of §13.22
Whittaker–Hill equation §28.31(i)
- applications §28.32(ii)
  - separation constants §28.32(ii)
Whittaker’s equation §13.14(i)
- fundamental solutions §13.14(v)
- numerically satisfactory solutions §13.14(v)
- relation to Kummer’s equation §13.14(i)
- standard solutions §13.14(i)
Wigner $\mathit{3j},\mathit{6j},\mathit{9j}$ symbols, see $\mathit{3j}$ symbols, $\mathit{6j}$ symbols, and $\mathit{9j}$ symbols.
Wilf–Zeilberger algorithm
- applied to generalized hypergeometric functions §16.4(iii)
Wilkinson’s polynomial §3.8(vi)
Wilson class orthogonal polynomials §18.25—§18.26(v)
- asymptotic approximations §18.26(v)
- definitions §18.25
- differences §18.26(iii)
- dualities §18.21(i)
- generating functions §18.26(iv)
- interrelations with other orthogonal polynomials Figure 18.21.1, Figure 18.21.1, Figure 18.21.1, §18.26(ii)—§18.26(ii)
- leading coefficients Table 18.25.2
- notation §18.1(ii)
- orthogonality properties Table 18.25.1
- relations to other functions
  - generalized hypergeometric functions §18.26(i), §18.26(iv)
  - $6j$ symbols §18.38(iii)
- standardizations §18.25(ii)—§18.25(iii)
- transformations of variable Table 18.25.1
- weight functions §18.25(ii)—§18.25(iii)
Wilson polynomials, see Wilson class orthogonal polynomials.
winding number
- of closed contour §1.9(iii)
WKB or WKBJ approximation, see Liouville–Green (or WKBJ) approximation.
Wronskian
- differential equations §1.13(i)
Wynn’s cross rule
- for Padé approximations §3.11(iv)
Wynn’s epsilon algorithm
- for sequences §3.9(iv)

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