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21: 33.1 Special Notation

§33.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►

Alternative Notations

►

$P_{\mathrm{\ell }}(ϵ,r)=(2\mathrm{\ell }+1)!f(ϵ,\mathrm{\ell };r)/2^{\mathrm{\ell }+1}$, $Q_{\mathrm{\ell }}(ϵ,r)=-(2\mathrm{\ell }+1)!h(ϵ,\mathrm{\ell };r)/(2^{\mathrm{\ell }+1}A(ϵ,\mathrm{\ell }))$.

►

$f^{(0)}(ϵ,\mathrm{\ell };r)=f(ϵ,\mathrm{\ell };r)$, $f(ϵ,\mathrm{\ell };r)=s(ϵ,\mathrm{\ell };r)$, $g(ϵ,\mathrm{\ell };r)=c(ϵ,\mathrm{\ell };r)$.

22: 34.1 Special Notation

§34.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►

$2j_1,2j_2,2j_3,2l_1,2l_2,2l_3$	nonnegative integers.
…

… ►

34.1.1

ⓘ

Symbols:

: $\mathit{3}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Sources:

Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))

Permalink:

http://dlmf.nist.gov/34.1.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

… ►For other notations for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

23: 31.1 Special Notation

§31.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►

$x$, $y$	real variables.
…

… ►These notations were introduced by Arscott in Ronveaux (1995, pp. 34–44). …

24: 30.1 Special Notation

§30.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). … ►

Other Notations

…

25: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ► ►Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ by ${\mathrm{P}}_{\nu }^{\mu }(x)$ and ${\mathrm{Q}}_{\nu }^{\mu }(x)$, respectively. Magnus et al. (1966) denotes ${𝖯}_{\nu }^{\mu }\left(x\right)$, ${𝖰}_{\nu }^{\mu }\left(x\right)$, $P_{\nu }^{\mu }\left(z\right)$, and $Q_{\nu }^{\mu }\left(z\right)$ by $P_{\nu }^{\mu }(x)$, $Q_{\nu }^{\mu }(x)$, ${𝔓}_{\nu }^{\mu }(z)$, and ${𝔔}_{\nu }^{\mu }(z)$, respectively. …

26: 20.1 Special Notation

§20.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►

Other Notations

… ►This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). … ►Additional notations that have been used in the literature are summarized in Whittaker and Watson (1927, p. 487).

27: 35.1 Special Notation

§35.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ► ►►

$a,b$	complex variables.
…

… ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

28: 15.1 Special Notation

§15.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►We use the following notations for the hypergeometric function: ►

15.1.1 $${}_2{}^{}F_1^{}(a,b;c;z)=F(a,b;c;z)=F(\genfrac{}{}{0pt}{}{a,b}{c};z),$$

ⓘ

Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.1.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.1 and Ch.15

…

29: 19.1 Special Notation

§19.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►All derivatives are denoted by differentials, not by primes. … ►This notation follows Byrd and Friedman (1971, 110). … ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. …

30: 29.1 Special Notation

§29.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►All derivatives are denoted by differentials, not by primes. … ►The notation for the eigenvalues and functions is due to Erdélyi et al. (1955, §15.5.1) and that for the polynomials is due to Arscott (1964b, §9.3.2). … ►Other notations that have been used are as follows: Ince (1940a) interchanges $a_{\nu }^{2m+1}\left(k^2\right)$ with $b_{\nu }^{2m+1}\left(k^2\right)$. …