# Legendre functions on the cut

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##### 1: 14.1 Special Notation
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 3: Bibliography O
• F. W. J. Olver and J. M. Smith (1983) Associated Legendre functions on the cut. J. Comput. Phys. 51 (3), pp. 502–518.
• ##### 4: Mathematical Introduction
Other examples are: (a) the notation for the Ferrers functions—also known as associated Legendre functions on the cut—for which existing notations can easily be confused with those for other associated Legendre functions14.1); (b) the spherical Bessel functions for which existing notations are unsymmetric and inelegant (§§10.47(i) and 10.47(ii)); and (c) elliptic integrals for which both Legendre’s forms and the more recent symmetric forms are treated fully (Chapter 19). …
##### 5: Bibliography S
• J. Segura and A. Gil (1999) Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.
• ##### 6: 14.27 Zeros
###### §14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …For all other values of the parameters $P^{\mu}_{\nu}\left(x\pm i0\right)$ has no zeros in the interval $(-\infty,-1)$. For complex zeros of $P^{\mu}_{\nu}\left(z\right)$ see Hobson (1931, §§233, 234, and 238).
##### 7: 14.22 Graphics
###### §14.22 Graphics
In the graphics shown in this section, height corresponds to the absolute value of the function and color to the phase. …
##### 8: 30.5 Functions of the Second Kind
30.5.3 $\mathsf{Qs}^{m}_{n}\left(x,0\right)=\mathsf{Q}^{m}_{n}\left(x\right);$
##### 9: 30.4 Functions of the First Kind
If $\gamma=0$, $\mathsf{Ps}^{m}_{n}\left(x,0\right)$ reduces to the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$:
30.4.2 $\mathsf{Ps}^{m}_{n}\left(x,0\right)=\mathsf{P}^{m}_{n}\left(x\right);$
##### 10: 30.8 Expansions in Series of Ferrers Functions
30.8.1 $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-R}^{\infty}(-1)^{k}a^{m}% _{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right),$
30.8.9 $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)=\sum_{k=-\infty}^{-N-1}(-1)^{k}{a% ^{\prime}}^{m}_{n,k}(\gamma^{2})\mathsf{P}^{m}_{n+2k}\left(x\right)+\sum_{k=-N% }^{\infty}(-1)^{k}a^{m}_{n,k}(\gamma^{2})\mathsf{Q}^{m}_{n+2k}\left(x\right),$