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Notations S

*ABCDEFGHIJKLMNOPQR♦S♦TUVWXYZ
𝔖 n
set of permutations of {1,2,,n}; §26.13
S n ( 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)
𝒮 n ( k ) = S ( n , k )
notation used by Fort (1948); §26.1
(with S(n,k): Stirling number of the second kind)
𝔖 n k = 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)
S 1 ( z ) = S ( 2 / π z )
alternative notation for the Fresnel integral; §7.1
(with S(z): Fresnel integral and π: the ratio of the circumference of a circle to its diameter)
S 2 ( z ) = S ( 2 z / π )
alternative notation for the Fresnel integral; §7.1
(with S(z): Fresnel integral and π: the ratio of the circumference of a circle to its diameter)
S μ , ν ( z )
Lommel function; (11.9.5)
s μ , ν ( z )
Lommel function; (11.9.3)
S n ( x )
dilated Chebyshev polynomial; (18.1.3)
𝒮 ( f ) ( s )
Stieltjes transform; (1.14.47)
s ( n , k )
Stirling number of the first kind; §26.8(i)
S ( n , k )
Stirling number of the second kind; §26.8(i)
S 1 ( 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)
S 2 ( 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)
S n m ( j ) ( z , γ )
radial spheroidal wave function; (30.11.3)
S m n ( 1 ) ( γ , x ) Ps n m ( x , γ 2 )
alternative notation for the spheroidal wave function of the first kind; §30.1
(with Psnm(x,γ2): spheroidal wave function of the first kind)
S m n ( 2 ) ( γ , x ) Qs n m ( x , γ 2 )
alternative notation for the spheroidal wave function of the second kind; §30.1
(with Qsnm(x,γ2): spheroidal wave function of the second kind)
S n m ( z , ξ )
Ince polynomials; §28.31(ii)
S n ( x ; q )
Stieltjes–Wigert polynomial; (18.27.18)
s ( ϵ , ; r )
regular Coulomb function; (33.14.9)
S ( k , h ) ( x )
Sinc function; §3.3(vi)
S n ( x ; a , b , c )
continuous dual Hahn polynomial; Table 18.25.1
sc ( z , k )
Jacobian elliptic function; (22.2.9)
scdE 2 n + 3 m ( z , k 2 )
Lamé polynomial; (29.12.8)
scE 2 n + 2 m ( z , k 2 )
Lamé polynomial; (29.12.5)
sd ( z , k )
Jacobian elliptic function; (22.2.7)
sdE 2 n + 2 m ( z , k 2 )
Lamé polynomial; (29.12.6)
Se n ( c , z ) = ce n ( z , q ) ce n ( 0 , q )
notation used by Stratton et al. (1941); §28.1
(with cen(z,q): Mathieu function)
Se n ( s , z ) = ce n ( z , q ) ce n ( 0 , q )
notation used by National Bureau of Standards (1967); §28.1
(with cen(z,q): Mathieu function)
se n ( z , q )
Mathieu function; §28.2(vi)
se ν ( z , q )
Mathieu function of noninteger order; (28.12.13)
Se ν ( z , q )
modified Mathieu function; (28.20.4)
sE 2 n + 1 m ( z , k 2 )
Lamé polynomial; (29.12.2)
sec z
secant function; (4.14.6)
sech z
hyperbolic secant function; (4.28.6)
seh n ( z , q ) = Se n ( 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)
σ n k = S ( n , k )
notation used by Moser and Wyman (1958b); §26.1
(with S(n,k): Stirling number of the second kind)
σ α ( n )
sum of powers of divisors of n; (27.2.10)
σ ( η )
Coulomb phase shift; (33.2.10)
σ n ( ν )
Rayleigh function; (10.21.55)
σ ( z | 𝕃 )
Weierstrass sigma function; §23.1
σ ( z ; g 2 , g 3 )
Weierstrass sigma function; §23.3(i)
sign x
sign of x; Common Notations and Definitions
sin z
sine function; (4.14.1)
Sin q ( x )
q-sine function; (17.3.4)
sin q ( x )
q-sine function; (17.3.3)
sinh z
hyperbolic sine function; (4.28.1)
sn ( z , k )
Jacobian elliptic function; (22.2.4)
sn ( z | m ) = sn ( z , m )
alternative notation; §22.1
(with sn(z,k): Jacobian elliptic function)
So n ( c , z ) = se n ( z , q ) se n ( 0 , q )
notation used by Stratton et al. (1941); §28.1
(with sen(z,q): Mathieu function)
So n ( s , z ) = se n ( z , q ) se n ( 0 , q )
notation used by National Bureau of Standards (1967); §28.1
(with sen(z,q): Mathieu function)
sup
least upper bound (supremum); Common Notations and Definitions