(For other notation see Notation for the Special Functions.)
$x$, $y$, $\tau $ | real variables. |
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$z=x+\mathrm{i}y$ | complex variable. |
$m$, $n$ | unless stated otherwise, nonnegative integers, used for order and degree, respectively. |
$\mu $, $\nu $ | general order and degree, respectively. |
$-\frac{1}{2}+\mathrm{i}\tau $ | complex degree, $\tau \in \mathbb{R}$. |
$\gamma $ | Euler’s constant (§5.2(ii)). |
$\delta $ | arbitrary small positive constant. |
$\psi \left(x\right)$ | logarithmic derivative of gamma function (§5.2(i)). |
${\psi}^{\prime}\left(x\right)$ | $d\psi \left(x\right)/dx$ . |
$\mathbf{F}(a,b;c;z)$ | Olver’s scaled hypergeometric function: $F(a,b;c;z)/\mathrm{\Gamma}\left(c\right)$. |
Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise.
The main functions treated in this chapter are the Legendre functions ${\U0001d5af}_{\nu}\left(x\right)$, ${\U0001d5b0}_{\nu}\left(x\right)$, ${P}_{\nu}\left(z\right)$, ${Q}_{\nu}\left(z\right)$; Ferrers functions ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$, ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions ${P}_{\nu}^{\mu}\left(z\right)$, ${Q}_{\nu}^{\mu}\left(z\right)$, ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$; conical functions ${\U0001d5af}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${\U0001d5b0}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${\widehat{\U0001d5b0}}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${P}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$, ${Q}_{-\frac{1}{2}+\mathrm{i}\tau}^{\mu}\left(x\right)$ (also known as Mehler functions).
Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$ and ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ by ${\mathrm{P}}_{\nu}^{\mu}(x)$ and ${\mathrm{Q}}_{\nu}^{\mu}(x)$, respectively. Magnus et al. (1966) denotes ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$, ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$, ${P}_{\nu}^{\mu}\left(z\right)$, and ${Q}_{\nu}^{\mu}\left(z\right)$ by ${P}_{\nu}^{\mu}(x)$, ${Q}_{\nu}^{\mu}(x)$, ${\U0001d513}_{\nu}^{\mu}(z)$, and ${\U0001d514}_{\nu}^{\mu}(z)$, respectively. Hobson (1931) denotes both ${\U0001d5af}_{\nu}^{\mu}\left(x\right)$ and ${P}_{\nu}^{\mu}\left(x\right)$ by ${P}_{\nu}^{\mu}\left(x\right)$; similarly for ${\U0001d5b0}_{\nu}^{\mu}\left(x\right)$ and ${Q}_{\nu}^{\mu}\left(x\right)$.