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14 Legendre and Related FunctionsNotation

§14.1 Special Notation

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

x, y, τ real variables.
z=x+iy complex variable.
m, n nonnegative integers used for order and degree, respectively.
μ, ν general order and degree, respectively.
-12+iτ complex degree, τ.
γ Euler’s constant (§5.2(ii)).
δ arbitrary small positive constant.
ψ(x) logarithmic derivative of gamma function (§5.2(i)).
ψ(x) dψ(x)/dx .
F(a,b;c;z) Olver’s scaled hypergeometric function: F(a,b;c;z)/Γ(c).

Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise.

The main functions treated in this chapter are the Legendre functions Pν(x), Qν(x), Pν(z), Qν(z); Ferrers functions Pνμ(x), Qνμ(x) (also known as the Legendre functions on the cut); associated Legendre functions Pνμ(z), Qνμ(z), Qνμ(z); conical functions P-12+iτμ(x), Q-12+iτμ(x), Q^-12+iτμ(x), P-12+iτμ(x), Q-12+iτμ(x) (also known as Mehler functions).

Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote Pνμ(x) and Qνμ(x) by Pνμ(x) and Qνμ(x), respectively. Magnus et al. (1966) denotes Pνμ(x), Qνμ(x), Pνμ(z), and Qνμ(z) by Pνμ(x), Qνμ(x), 𝔓νμ(z), and 𝔔νμ(z), respectively. Hobson (1931) denotes both Pνμ(x) and Pνμ(x) by Pνμ(x); similarly for Qνμ(x) and Qνμ(x).