Index W
-
Waring’s problem
- number theory §27.13(iii)
- water waves
- Watson integrals
-
Watson’s sum
- Andrews’ terminating -analog ¶ ‣ §17.7(iii)
- Gasper–Rahman -analog ¶ ‣ §17.7(iii)
-
Watson’s expansions
- theta functions §20.8
-
Watson’s identities
- theta functions §20.7(v)
- Watson’s lemma
-
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 §28.33(ii), §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–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 Figure 23.4.6, Figure 23.4.6, §23.4(i), §23.5(v)
- Fourier series §23.8(i)
- graphics
- 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 Figure 23.4.7, Figure 23.4.7, §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)
- roots §23.3(i)
- Laurent series §23.9
- lemniscatic case Figure 23.4.7, Figure 23.4.7, §23.4(i), §23.5(iii)
- notation §23.1
- -tuple formulas §23.10(iii)
- 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)
- series of cosecants or cotangents §23.8(ii)
- tables §23.23
- zeros §23.13, §23.2(ii)
- Weierstrass -test, see -test for uniform convergence.
- Weierstrass -function, see Weierstrass elliptic functions.
- Weierstrass product ¶ ‣ §1.10(ix)
- Weierstrass sigma function, see Weierstrass elliptic functions.
- Weierstrass zeta function, see Weierstrass elliptic functions.
- weighted means §1.2(iv)
-
weight functions
- cubature §3.5(x)
- definition ¶ ‣ §18.2(i), §3.5(iv)
- Freud §18.32
- least squares approximations §3.11(v)
- logarithmic ¶ ‣ §3.5(v)
- minimax rational approximations §3.11(iii)
- quadrature §3.5(iv), §3.5(v)
-
Weniger’s transformation
- for sequences §3.9(v)
-
Whipple’s sum
- Gasper–Rahman -analog ¶ ‣ §17.7(iii)
- Whipple’s formula
-
Whipple’s sum
- generalized hypergeometric functions ¶ ‣ §16.4(ii)
-
Whipple’s theorem
- Watson’s -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)
- analytical properties ¶ ‣ §13.14(i)
- analytic continuation §13.14(ii)
-
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 and/or §13.20(iv)
- large §13.21, §13.21(iv)
- large §13.20(i), §13.20(v)
- uniform §13.20(i), §13.21(iv)
- asymptotic expansions for large argument §13.19, §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)
-
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)
- integral transforms in terms of §13.23(iv)
- interrelations §13.14(vii)
- large argument §2.11(vi)
-
limiting forms
- as §13.14(iii)
- as §13.14(iv)
- 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
-
Whittaker–Hill equation §28.31(i)
-
applications §28.32(ii)
- separation constants §28.32(ii)
-
applications §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 symbols, see symbols, symbols, and 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, ¶ ‣ §18.26(ii), §18.26(ii)
- leading coefficients Table 18.25.2
- normalizations ¶ ‣ §18.25(iii), §18.25(ii)
- notation ¶ ‣ §18.1(ii)
- orthogonality properties Table 18.25.1
- relation to generalized hypergeometric functions §18.26(i), §18.26(iv)
- transformations of variable Table 18.25.1
- weight functions ¶ ‣ §18.25(iii), §18.25(ii)
- 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)