14 Legendre and Related Functions Complex Arguments 14.22 Graphics 14.24 Analytic Continuation

§14.23 Values on the Cut

Notes:: For (14.23.1) and (14.23.5) see Olver (1997b, p. 185), and for (14.23.7) see Dunster (1991). (14.23.2) may be derived from (14.23.4) and (14.24.2). (14.23.3) may be derived from (14.3.11), (14.3.12), and (14.9.14). (14.23.4) may be derived from (14.23.1). (14.23.6) may be derived from (14.23.1) and (14.23.2).
Keywords:: Ferrers functions, analytic continuation, associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.23

When ,

14.23.1 $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right)=e^{{\mp\mu\pi i/2}% }\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.23
Permalink:: http://dlmf.nist.gov/14.23.E1
Encodings:: TeX, pMML, png

14.23.2 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right)=\frac% {e^{{\pm\mu\pi i/2}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)}\left% (\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\mp\tfrac{1}{2}% \pi i\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.23
Permalink:: http://dlmf.nist.gov/14.23.E2
Encodings:: TeX, pMML, png

In terms of the hypergeometric function $\mathop{\mathbf{F}\/}\nolimits$ (§14.3(i))

14.23.3 $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right)=\frac% {e^{{\mp\nu\pi i/2}}\pi^{{3/2}}\left(1-x^{2}\right)^{{\mu/2}}}{2^{{\nu+1}}}% \left(\frac{x\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{1}{2}\mu-\frac{1}{2}% \nu+\frac{1}{2},\frac{1}{2}\nu+\frac{1}{2}\mu+1;\frac{3}{2};x^{2}\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+\frac{1}{2}% \right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1% }{2}\right)}\mp i\frac{\mathop{\mathbf{F}\/}\nolimits\!\left(\frac{1}{2}\mu-% \frac{1}{2}\nu,\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2};\frac{1}{2};x^{2}% \right)}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu-\frac{1}{2}\mu+1% \right)\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}\nu+\frac{1}{2}\mu+1\right% )}\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, $\mathop{\mathbf{F}\/}\nolimits\!\left(a,b;c;z\right)$ : Olver’s hypergeometric function, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.23
Permalink:: http://dlmf.nist.gov/14.23.E3
Encodings:: TeX, pMML, png

Conversely,

14.23.4 $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=e^{{\pm\mu\pi i% /2}}\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right),$

Symbols:: $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : base of exponential function, : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.3.2 (corrected)
Referenced by:: §14.23, §14.23, §14.3(iv)
Permalink:: http://dlmf.nist.gov/14.23.E4
Encodings:: TeX, pMML, png

14.23.5 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=\tfrac{1}{2}% \mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\left(e^{{-\mu\pi i/2}}% \mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x+i0\right)+e^{{\mu% \pi i/2}}\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x-i0\right)% \right),$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, : real variable, $\mu$ : general order and $\nu$ : general degree
A&S Ref:: 8.3.4 (modified)
Referenced by:: §14.23
Permalink:: http://dlmf.nist.gov/14.23.E5
Encodings:: TeX, pMML, png

or equivalently,

14.23.6 $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)=e^{{\mp\mu\pi i% /2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\mu+1\right)\mathop{\boldsymbol{Q}^{% {\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right)\pm\tfrac{1}{2}\pi ie^{{\pm\mu% \pi i/2}}\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\pm i0\right).$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the second kind, $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\boldsymbol{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : Olver’s associated Legendre function, : base of exponential function, : real variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.23, §14.23
Permalink:: http://dlmf.nist.gov/14.23.E6
Encodings:: TeX, pMML, png

If cuts are introduced along the intervals $(-\infty,-1]$ and $[1,\infty)$ , then (14.23.4) and (14.23.6) could be used to extend the definitions of $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathsf{Q}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ to complex .

The conical function defined by (14.20.2) can be represented similarly by

14.23.7 $\mathop{\widehat{\mathsf{Q}}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!% \left(x\right)=\tfrac{1}{2}e^{{3\mu\pi i/2}}\mathop{Q^{{-\mu}}_{{-\frac{1}{2}+% i\tau}}\/}\nolimits\!\left(x-i0\right)+\tfrac{1}{2}e^{{-3\mu\pi i/2}}\mathop{Q% ^{{-\mu}}_{{-\frac{1}{2}-i\tau}}\/}\nolimits\!\left(x+i0\right).$

Symbols:: $\mathop{\widehat{\mathsf{Q}}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!% \left(x\right)$ : conical function, $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : base of exponential function, : real variable, $\tau$ : real variable and $\mu$ : general order
Referenced by:: §14.23
Permalink:: http://dlmf.nist.gov/14.23.E7
Encodings:: TeX, pMML, png

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