Index S

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

$S$ -matrix scattering
- Coulomb functions §33.22(v)
saddle points §2.4(iv)
- coalescing §2.4(v), §36.12—§36.12(iii)
sampling expansions
- parabolic cylinder functions §12.16
scaled gamma function §8.18(ii)
scaled Riemann theta functions
- definition §21.2(i)
scaled spheroidal wave functions §30.15—§30.15(v)
- bandlimited §30.15(iii)
- extremal properties §30.15(v)
- Fourier transform §30.15(iii)
- integral equation §30.15(ii)
- orthogonality §30.15(iv)
scaling laws
- for diffraction catastrophes §36.6
scattering problems
- associated Legendre functions §14.31(iii)
- Coulomb functions Ch.33—§33.22(vii)
scattering theory
- Mathieu functions 4th item
Schläfli-type integrals
- Kelvin functions §10.64
Schläfli–Sommerfeld integrals
- Bessel and Hankel functions §10.9(ii)—§10.9(ii)
Schläfli’s integrals
- Bessel functions §10.9(i), §10.9(ii)
Schottky group
- Riemann surface 1st item
Schottky problem
- Riemann surface §21.9
Schröder numbers
- definition §26.6(i)
- generating function §26.6(ii)
- relation to lattice paths §26.6(i)
- table Table 26.6.4
Schrödinger equation
- Airy functions §9.16
- Coulomb functions §33.22(i)—§33.22(vii)
- nonlinear
  - Jacobian elliptic functions §22.19(iii)
  - Riemann theta functions §21.9
- $q$ -deformed quantum mechanical §17.17
- solutions in terms of classical orthogonal polynomials §18.39(i)—§18.39(i)
- theta functions §20.13
Schwarz reflection principle §1.10(ii)
Schwarz’s lemma §1.10(v)
Schwarzian derivative §1.13(iv)
Scorer functions §9.12
- analytic properties §9.12(i)
- applications §9.16
- approximations
  - expansions in Chebyshev series §9.19(iv)
- asymptotic expansions §9.12(viii)
- computation §9.17(ii), §9.17(iii)
- computation by quadrature §3.5(ix)
- connection formulas §9.12(v)
- definition §9.12(i)
- differential equation §9.12(i)
  - initial values §9.12(iii)
  - numerically satisfactory solutions §9.12(iv)
  - standard solutions §9.12(i)
- graphs §9.12(ii)
- integral representations §9.12(i), §9.12(vii), §9.12(vii)—§9.12(vii)
- integrals
  - asymptotic expansions §9.12(viii)
  - tables 1st item
- Maclaurin series §9.12(vi)
- notation §9.1
- tables 2nd item, 3rd item, §9.18(vi)
- zeros §9.12(ix)
  - computation §9.17(v)
  - tables 4th item
secant function, see trigonometric functions.
sectorial harmonics §14.30(i)
Selberg integrals
- generalized elliptic integrals §19.35(i)
Selberg-type integrals
- gamma function §5.14
separable Gauss sum
- number theory §27.10
Shanks’ transformation
- for sequences §3.9(iv)
ship wave §36.13—§36.13
sieve of Eratosthenes
- prime numbers §27.18
sigma function, see Weierstrass elliptic functions.
signal analysis
- spheroidal wave functions §30.15—§30.15(v)
simple closed contour §1.9(iii)
simple closed curve §1.6(iv)
simple discontinuity §1.4(ii)
simple zero §1.10(i)
simply-connected domain §1.13(i)
Sinc function §3.3(vi)
sine function, see trigonometric functions.
sine integrals §6.2(ii)
- applications
  - Gibbs phenomenon §6.16(i)
  - physical §6.17
- approximations 4th item
- asymptotic expansions §6.12(i)
  - exponentially-improved §6.12(ii)
- auxiliary functions, see auxiliary functions for sine and cosine integrals.
- Chebyshev-series expansions 2nd item—§6.20(ii)
- computation §6.18
- definition §6.2(ii)
- expansion in spherical Bessel functions §6.10(ii)
- generalized §8.21—§8.21(viii)
- graphics Figure 6.3.2, Figure 6.3.2, Figure 6.3.2
- hyperbolic analog §6.2(ii)
- integral representations §6.7(ii)
- integrals §6.14(i), §6.14(ii)
- Laplace transform §6.14(i)
- maxima and minima §6.16(i)
- notation §6.1
- power series §6.6
- relations to exponential integrals §6.5
- sums §6.15
- tables 1st item, 2nd item
- value at infinity §6.2(ii)
- zeros §6.13
  - asymptotic expansion §6.13
  - computation §6.18(iii)
sine-Gordon equation
- Jacobian elliptic functions §22.19(iii)
- Painlevé transcendents §32.13(ii)
singularities
- movable §32.2(i)
singularity
- branch point §1.10(vi)
- essential §1.10(iii)
- isolated §1.10(iii)
- isolated essential §1.10(iii)
- pole §1.10(iii)
- removable §1.10(iii), §1.4(ii)
$\mathit{6j}$ symbols §34.4
- addition theorem §34.5(vi)
- applications §34.12
- approximations for large parameters §34.8
- computation §34.13
- definition §34.4
  - alternative §34.5(vi)
- generating functions §34.5(v)
- graphical method §34.9
- notation §34.1
- orthogonality §34.5(iv)
- recursion relations §34.5(iii)
- Regge symmetries §34.5(ii)
- representation as
  - finite sum of algebraic quantities §34.5(i)
  - finite sum of $\mathit{3j}$ symbols §34.4
  - generalized hypergeometric functions §34.4
- special cases §34.5(i)
- sum rules §34.5(vi)
- summation convention §34.3(iv)
- sums §34.5(vi)
- symmetry §34.5(ii)
- tables §34.14
- zeros §34.10
SL $(2,\mathbb{Z})$ bilinear transformation §23.15(i)
Sobolev polynomials §18.36(ii)
soliton theory
- classical orthogonal polynomials §18.38(ii)
solitons
- Jacobian elliptic functions §22.19(iii)
- Weierstrass elliptic functions §23.21(ii)
spatio-temporal dynamics
- Heun functions §31.14(ii)
special distributions
- Fourier transforms §1.16(viii)
spectral problems
- Heun’s equation §31.17(i)
- separation of variables §31.17(i)
spherical Bessel functions Ch.10
- addition theorems §10.60(i)
- analytic properties §10.47(ii)
- applications
  - electromagnetic scattering §10.73(ii)
  - Helmholtz equation §10.73(ii)
  - wave equation §10.73(ii)
- approximations §10.76(iii)
- asymptotic approximations for large order, see uniform asymptotic expansions for large order
- computation Ch.10—§10.74(v)
- continued fractions §10.55
- cross-products §10.50
- definitions §10.47(ii), §10.47(ii)
- derivatives §10.51(i)—§10.51(ii)
  - zeros §10.58, 2nd item
- differential equations §10.47(i)
  - numerically satisfactory solutions §10.47(iii)
  - singularities §10.47(i)
  - standard solutions §10.47(ii)
- Dirac delta §1.17(ii)
- duplication formulas §10.60(ii)
- explicit formulas
  - modified functions §10.49(ii)
  - sums or differences of squares §10.49(iv)
  - unmodified functions §10.49(i)—§10.49(i)
- generating functions §10.56
- graphs §10.48—§10.48
- integral representations §10.54
- integrals §10.59
  - computation §10.74(vii)
- interrelations §10.47(iv)
- limiting forms §10.52
- modified §10.47(ii)
- notation §10.1
- of the first, second, and third kinds §10.47(ii)
- power series §10.53
- Rayleigh’s formulas §10.49(iii)
- recurrence relations §10.51(i)—§10.51(ii)
- reflection formulas §10.47(v)
- sums §10.60
  - addition theorems §10.60(i)
  - compendia §10.60(iv)
  - duplication formulas §10.60(ii)
- tables §10.75(ix)—§10.75(ix)
- uniform asymptotic expansions for large order §10.57
- Wronskians §10.50
- zeros §10.58
spherical Bessel transform §10.74(vii)
- computation §10.74(vii)
spherical coordinates §1.5(ii)
spherical harmonics §14.30(i)
- addition theorem §14.30(iii)
- applications §14.30(iv)
- basic properties §14.30(ii)—§14.30(ii)
- definitions §14.30(i)
- Dirac delta §1.17(iii)
- distributional completeness §14.30(iii)
- Lamé polynomials §29.18(iii)
- relation to $\mathit{3j}$ symbols §34.3(vii)
- sums §14.30(iii)
- zonal §18.38(ii)
spherical polar coordinates, see spherical coordinates.
spherical triangles
- solution of §4.42(iii)
spherical trigonometry
- Jacobian elliptic functions §22.18(iii)
sphero-conal coordinates §29.18(i)
spheroidal coordinates, see oblate spheroidal coordinates and prolate spheroidal coordinates.
spheroidal differential equation §30.2(i)
- eigenvalues §30.3—§30.3(iv)
  - asymptotic behavior §30.9—§30.9(iii)
  - computation §30.16(i)
  - continued-fraction equation §30.3(iii)
  - graphics §30.7(i)—§30.7(i)
  - power-series expansion §30.3(iv)
  - tables §30.17
- Liouville normal form §30.2(ii)
- singularities §30.2(i)
- special cases §30.2(iii)
- with complex parameter §30.6
spheroidal harmonics
- oblate §14.30(i)
- prolate §14.30(i)
spheroidal wave functions §30.1
- addition theorem §30.10
- applications
  - signal analysis §30.15—§30.15(v)
  - wave equation §30.13—§30.14(v)
- approximations §30.9(iii)
- as confluent Heun functions §31.12
- asymptotic behavior
  - as $x\to\pm 1$ §30.9(iii)
  - for large $\left|\gamma^{2}\right|$ §30.9(i)—§30.9(iii)
- computation §30.16(ii)—§30.16(ii)
- convolutions §30.10
- Coulomb §30.12
- definitions §30.4(i), §30.5
- differential equation §30.2(i)
- eigenvalues §30.3
- elementary properties §30.4(ii)
- expansions in series of Ferrers functions §30.8—§30.8(ii)
  - asymptotic behavior of coefficients §30.8(i)
  - tables of coefficients §30.17
- expansions in series of spherical Bessel functions §30.10
- Fourier transform §30.15(iii)
- generalized §30.12
- graphics §30.7(ii)—§30.7(iv)
- integral equations §30.10, §30.15(ii)
- integrals §30.10
- notation §30.1
- oblate angular §30.4(i)
- of complex argument §30.6
- of the first kind §30.4
- of the second kind §30.5
- orthogonality §30.4(iv)
- other notations §30.1
- power-series expansions §30.3(iv)
- products §30.10
- prolate angular §30.4(i)
- radial §30.11—§30.11(ii)
- scaled §30.15(i)
- tables §30.17
- with complex parameters §30.6
- zeros §30.4(ii)

spline functions
- Bernoulli monosplines §24.17(ii)
- cardinal monosplines §24.17(ii)
- cardinal splines §24.17(ii)
- Euler splines §24.17(ii)
splines
- Bézier curves §3.11(vi)
- definitions §3.11(vi)
square-integrable function §1.4(v)
stability problems
- Mathieu functions §28.33(iii)
stable polynomials §1.11(v)
- Hurwitz criterion §1.11(v)
statistical analysis
- multivariate
  - functions of matrix argument §35.9
statistical applications
- functions of matrix argument §35.9
statistical mechanics
- application to combinatorics §26.20
- Heun functions §31.17(ii)
- incomplete beta functions §8.24(ii)
- Jacobian elliptic functions §22.18(iii)
- modular functions §23.21
- $q$ -hypergeometric function §17.17
- solvable models §5.20
- theta functions §20.12(ii)
statistical physics
- Bernoulli and Euler polynomials §24.18
- Painlevé transcendents §32.16
Steed’s algorithm
- for continued fractions §3.10(iii)
steepest-descent paths
- numerical integration §3.5(ix)—§3.5(ix)
Stickelberger codes
- Bernoulli numbers 2nd item
Stieltjes fraction ( $S$ -fraction) §3.10(ii)
Stieltjes polynomials
- definition §31.15(i)
- orthogonality §31.15(iii)
- products §31.15(iii)
- zeros §31.15(ii)
  - electrostatic interpretation §31.15(ii)
Stieltjes transform
- analyticity §1.14(vi)
- asymptotic expansions §2.6(ii)—§2.6(ii)
- convergence §1.14(vi)
- definition §1.14(vi), §2.6(ii)
- derivatives §1.14(vi)
- generalized §2.6(ii)
- inversion §1.14(vi)
- representation as double Laplace transform §1.14(vi)
Stieltjes–Wigert polynomials §18.27(vi)
- asymptotic approximations §18.29
Stirling cycle numbers §26.13
Stirling numbers (first and second kinds)
- asymptotic approximations §26.8(vii)
- definitions §26.8(i)
- generalized §26.8(vii)
- generating functions §26.8(ii)
- identities §26.8(v)
- notations §26.1
- recurrence relations §26.8(iv)
- relations to Bernoulli numbers §24.15(iii)
- special values §26.8(iii)
- tables §26.21, Table 26.8.1, Table 26.8.2
Stirling’s formula §5.11(i)
Stirling’s series §5.11(i)
Stokes line §2.11(iv)
Stokes multipliers §2.7(ii)
Stokes phenomenon §2.11(iv)
- complementary error function §7.12(i)
- incomplete gamma functions §8.22(i)
- smoothing of §2.11(iv)
Stokes sets §36.5(i)—§36.5(iv)
- cuspoids §36.5(ii)
- definitions §36.5(i)
- umbilics §36.5(iii)
- visualizations §36.5(iv)—§36.5(iv)
Stokes’ theorem for vector-valued functions §1.6(v)
string theory
- beta function §5.20
- elliptic integrals §19.35(ii)
- modular functions §23.21
- Painlevé transcendents §32.16
- Riemann theta functions §21.9
- theta functions §20.12(ii)
Struve functions, see Struve functions and modified Struve functions.
Struve functions and modified Struve functions Ch.11
- analytic continuation §11.4(iii)
- applications
  - physical §11.12—§11.12
- approximations §11.15(i)
- argument $xe^{\pm 3\pi i/4}$ §11.8
- asymptotic expansions
  - generalized §11.6(ii)
  - large argument §11.6(i)
  - large order §11.6(ii)
  - remainder terms §11.6(i)
- computation §11.13(i)
- definitions §11.2
- derivatives §11.4(v)
  - with respect to order §11.4(vi)
- differential equations §11.2(ii)—§11.2(iii)
  - numerically satisfactory solutions §11.2(iii)—§11.2(iii)
  - particular solutions §11.2(ii), §11.2(ii)
- graphics Figure 11.3.1, Figure 11.3.1, Figure 11.3.1, Figure 11.3.10, Figure 11.3.10, Figure 11.3.10, Figure 11.3.11, Figure 11.3.11, Figure 11.3.11, Figure 11.3.12, Figure 11.3.12, Figure 11.3.12, Figure 11.3.13, Figure 11.3.13, Figure 11.3.13, Figure 11.3.14, Figure 11.3.14, Figure 11.3.14, Figure 11.3.15, Figure 11.3.15, Figure 11.3.15, Figure 11.3.16, Figure 11.3.16, Figure 11.3.16, Figure 11.3.17, Figure 11.3.17, Figure 11.3.17, Figure 11.3.18, Figure 11.3.18, Figure 11.3.18, Figure 11.3.19, Figure 11.3.19, Figure 11.3.19, Figure 11.3.2, Figure 11.3.2, Figure 11.3.2, Figure 11.3.20, Figure 11.3.20, Figure 11.3.20, Figure 11.3.3, Figure 11.3.3, Figure 11.3.3, Figure 11.3.4, Figure 11.3.4, Figure 11.3.4, Figure 11.3.5, Figure 11.3.5, Figure 11.3.5, Figure 11.3.6, Figure 11.3.6, Figure 11.3.6, Figure 11.3.7, Figure 11.3.7, Figure 11.3.7, Figure 11.3.8, Figure 11.3.8, Figure 11.3.8, Figure 11.3.9, Figure 11.3.9, Figure 11.3.9
- half-integer orders §11.4(i)
- incomplete §11.14(v)
- inequalities §11.4(ii)
- integral representations
  - along real line §11.5(i)
  - compendia §11.5(iii)
  - contour integrals §11.5(ii)
  - Mellin–Barnes type §11.5(ii)
- integrals
  - compendia §11.7(v)
  - definite §11.7(ii)
  - indefinite §11.7(i)—§11.7(i)
  - Laplace transforms §11.7(iii)
  - products §11.7(ii)
  - tables §11.14(iii)
  - with respect to order §11.7(iv)
- Kelvin-function analogs §11.8
- notation §11.1
- order §11.1
- power series §11.2(i)
- principal values §11.2(i)
- recurrence relations §11.4(v)
- relations to Anger–Weber functions §11.10(vi)
- series expansions
  - Bessel functions §11.4(iv)
  - Chebyshev §11.15(i)
  - power series §11.2(i)
- sums §11.7(v)
- tables §11.14(ii)
- zeros §11.4(vii)
Struve’s equation, see Struve functions and modified Struve functions, differential equations.
Sturm–Liouville eigenvalue problems
- ordinary differential equations §3.7(iv)
summability methods for integrals
- Abel §1.15(iv)
- Cesàro §1.15(iv)
- Fourier integrals
  - conjugate Poisson integral §1.15(v)
  - Fejér kernel §1.15(v)
  - Poisson integral §1.15(v)
  - Poisson kernel §1.15(v)
- fractional derivatives §1.15(vii)
- fractional integrals §1.15(vi)
summability methods for series
- Abel §1.15(i)
- Borel §1.15(i)
- Cesàro §1.15(i)
  - general §1.15(i)
- convergence §1.15(ii)
- Fourier series
  - Abel means §1.15(iii)
  - Cesàro means §1.15(iii)
  - Fejér kernel §1.15(iii)
  - Poisson kernel §1.15(iii)
- regular §1.15(ii)
- Tauberian theorems §1.15(viii)
summation by parts §2.10(ii)
summation formulas
- Boole §24.17(i)
- Euler–Maclaurin §24.17(i)
sums of powers
- as Bernoulli or Euler polynomials §24.4(iii)
- tables §24.20
supersonic flow
- Lamé polynomials §29.19(ii)
support
- of a function §1.16(i)
surface, see parametrized surfaces.
surface harmonics of the first kind §14.30(i)
surface-wave problems
- Struve functions §11.12
swallowtail bifurcation set
- formula §36.4(i)
- picture §36.4(ii)
swallowtail canonical integral §36.2(i)
- asymptotic approximations §36.11—§36.12(iii)
- convergent series §36.8
- differential equations §36.10(ii)
- formulas for Stokes set §36.5(ii)
- integral identities §36.9
- picture of Stokes set §36.5(iv)
- pictures of modulus Figure 36.3.2, Figure 36.3.2, Figure 36.3.2, Figure 36.3.3, Figure 36.3.3, Figure 36.3.3, Figure 36.3.4, Figure 36.3.4, Figure 36.3.4, Figure 36.3.5, Figure 36.3.5, Figure 36.3.5
- pictures of phase Figure 36.3.14, Figure 36.3.14
- scaling laws §36.6
- zeros §36.7(iv)
swallowtail catastrophe §36.2(i), Figure 36.5.2, Figure 36.5.2, Figure 36.5.3, Figure 36.5.3, Figure 36.5.4, Figure 36.5.4, Figure 36.5.7, Figure 36.5.7
symmetric elliptic integrals §19.16(i)
- addition theorems §19.26—§19.26(iii)
- advantages of symmetry §19.15—§19.15
- applications
  - mathematical Ch.19—§19.35(i)
  - physical §19.33(ii)—§19.35(ii)
  - statistical §19.31
- arithmetic-geometric mean §19.22(ii)
- asymptotic approximations and expansions §19.27—§19.27(vi), §2.6(ii)
- Bartky’s transformation §19.22(i)
- change of parameter of $R_{J}$ §19.21(iii)
- circular cases §19.20(iii)—§19.20(iii), §19.21(iii)
- complete §19.1
- computation §19.36—§19.38
- connection formulas §19.21
- degree §19.16(ii)
- derivatives §19.18(i)
- differential equations §19.18(ii), §19.18(ii)
- duplication formulas §19.26(iii)
- elliptic cases of $R_{-a}(\mathbf{b};\mathbf{z})$ §19.16(iii)
- first, second, and third kinds §19.1
- Gauss transformations §19.15, §19.22(iii)—§19.22(iii)
- general lemniscatic case §19.20(i), §19.20(iv)
- graphics §19.17—§19.17
- hyperbolic cases §19.20(iii)—§19.20(iii), §19.21(iii)
- inequalities
  - complete integrals §19.24(i)—§19.24(i)
  - incomplete integrals §19.24(ii)
- integral representations §19.23
- integrals of §19.28—§19.28
- Landen transformations §19.15, §19.22(iii)—§19.22(iii)
- notation §19.1
- permutation symmetry §19.15, §19.16(ii)
- power-series expansions §19.19—§19.19
- reduction of general elliptic integrals §19.29—§19.29(iii)
- relations to other functions
  - Appell functions §19.25(vii)
  - Bulirsch’s elliptic integrals §19.25(ii)
  - hypergeometric function §19.25(vii)
  - Jacobian elliptic functions §19.25(v)—§19.25(v)
  - Lauricella’s function §19.25(vii)
  - Legendre’s elliptic integrals §19.25(i), §19.25(iii)
  - theta functions §19.25(iv)
  - Weierstrass elliptic functions §19.25(vi)
- special cases §19.20—§19.20(v)
- tables §19.37(iv)
- transformations replaced by symmetry §19.15, §19.22(iii), §19.25(i)
symmetries
- of canonical integrals §36.2(iii)
Szegő–Askey polynomials §18.33(iv)
Szegő–Szász inequality
- Jacobi polynomials §18.14(iii)