Digital Library of Mathematical Functions
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Notations

Notations S

*ABCDEFGHIJKLMNOPQR♦S♦TUVWXYZ
𝔖n
set of permutations of {1,2,,n}; 26.13
𝒮n(k)=S(n,k)
notation used by Fort (1948); 26.1
(with S(n,k): Stirling number of the second kind)
Sn(k)=s(n,k)
notation used by Abramowitz and Stegun (1964, Chapter 24), Fort (1948); 26.1
(with s(n,k): Stirling number of the first kind)
S1(n-1,n-k)=s(n,k)/(-1)n-k
notation used by Carlitz (1960), Gould (1960); 26.1
(with s(n,k): Stirling number of the first kind)
Snk=s(n,k)
notation used by (Jordan, 1939), Moser and Wyman (1958a); 26.1
(with s(n,k): Stirling number of the first kind)
𝔖nk=S(n,k)
notation used by Jordan (1939); 26.1
(with S(n,k): Stirling number of the second kind)
S(z)
Fresnel integral; (7.2.8)
S1(z)=S(2/πz)
alternative notation for the Fresnel integral; 7.1
(with S(z): Fresnel integral)
S2(z)=S(2z/π)
alternative notation for the Fresnel integral; 7.1
(with S(z): Fresnel integral)
Sn(x)
dilated Chebyshev polynomial; (18.1.3)
Sμ,ν(z)
Lommel function; (11.9.5)
sμ,ν(z)
Lommel function; (11.9.3)
Snm(j)(z,γ)
radial spheroidal wave function; (30.11.3)
𝒮(f;s)
Stieltjes transform; (1.14.47)
S(n,k)
Stirling number of the second kind; 26.8(i)
s(n,k)
Stirling number of the first kind; 26.8(i)
S2(k,n-k)=S(n,k)
notation used by Carlitz (1960), Gould (1960); 26.1
(with S(n,k): Stirling number of the second kind)
Sn(x;q)
Stieltjes–Wigert polynomial; (18.27.18)
Smn(1)(γ,x)Psnm(x,γ2)
alternative notation for the spheroidal wave function of the first kind; 30.1
(with Psnm(x,γ2): spheroidal wave function of the first kind)
Smn(2)(γ,x)Qsnm(x,γ2)
alternative notation for the spheroidal wave function of the second kind; 30.1
(with Qsnm(x,γ2): spheroidal wave function of the second kind)
Snm(z,ξ)
Ince polynomials; 28.31(ii)
s(ϵ,;r)
regular Coulomb function; (33.14.9)
S(k,h)(x)
Sinc function; 3.3(vi)
Sn(x;a,b,c)
continuous dual Hahn polynomial; 18.25.1
sc(z,k)
Jacobian elliptic function; (22.2.9)
scdE2n+3m(z,k2)
Lamé polynomial; (29.12.8)
scE2n+2m(z,k2)
Lamé polynomial; (29.12.5)
sd(z,k)
Jacobian elliptic function; (22.2.7)
sdE2n+2m(z,k2)
Lamé polynomial; (29.12.6)
Seν(z,q)
modified Mathieu function; (28.20.4)
Sen(s,z)=cen(z,q)cen(0,q)
notation used by National Bureau of Standards (1967); 28.1
(with cen(z,q): Mathieu function)
Sen(c,z)=cen(z,q)cen(0,q)
notation used by Stratton et al. (1941); 28.1
(with cen(z,q): Mathieu function)
seν(z,q)
Mathieu function of noninteger order; (28.12.13)
sen(z,q)
Mathieu function; 28.2(vi)
sE2n+1m(z,k2)
Lamé polynomial; (29.12.2)
secz
secant function; (4.14.6)
sechz
hyperbolic secant function; (4.28.6)
sehn(z,q)=Sen(z,q)
notation used by Campbell (1955); 28.1
(with Seν(z,q): modified Mathieu function)
Shi(z)
hyperbolic sine integral; (6.2.15)
Si(z)
sine integral; (6.2.9)
si(z)
sine integral; (6.2.10)
Si(a,z)
generalized sine integral; (8.21.2)
si(a,z)
generalized sine integral; (8.21.1)
σnk=S(n,k)
notation used by Moser and Wyman (1958b); 26.1
(with S(n,k): Stirling number of the second kind)
σn(ν)
Rayleigh function; (10.21.55)
σ(η)
Coulomb phase shift; (33.2.10)
σα(n)
sum of powers of divisors of n; (27.2.10)
σ(z|𝕃)
Weierstrass sigma function; 23.1
σ(z) (= σ(z|𝕃) = σ(z;g2,g3))
Weierstrass sigma function; (23.2.6)
σ(z;g2,g3)
Weierstrass sigma function; 23.3(i)
signx
sign of x; Common Notations and Definitions
sinz
sine function; (4.14.1)
Sinq(x)
q-sine function; (17.3.4)
sinq(x)
q-sine function; (17.3.3)
sinhz
hyperbolic sine function; (4.28.1)
sn(z|m)=sn(z,m)
alternative notation; 22.1
(with sn(z,k): Jacobian elliptic function)
sn(z,k)
Jacobian elliptic function; (22.2.4)
Son(s,z)=sen(z,q)sen(0,q)
notation used by National Bureau of Standards (1967); 28.1
(with sen(z,q): Mathieu function)
Son(c,z)=sen(z,q)sen(0,q)
notation used by Stratton et al. (1941); 28.1
(with sen(z,q): Mathieu function)
sup
least upper bound (supremum); Common Notations and Definitions