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1: 14.31 Other Applications

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§14.31(iii) Miscellaneous

… ►Legendre functions $P_{\nu }\left(x\right)$ of complex degree $\nu$ appear in the application of complex angular momentum techniques to atomic and molecular scattering (Connor and Mackay (1979)). …

2: 14.1 Special Notation

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$x$, $y$, $\tau$	real variables.
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$-\frac{1}{2}+\mathrm{i}\tau$	complex degree, $\tau \in ℝ$.
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3: 18.10 Integral Representations

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18.10.8 $$p_n(x)=\frac{g_0(x)}{2\pi \mathrm{i}}{\int }_C{\left(g_1(z,x)\right)}^n{g}_2(z,x){(z-c)}^{-1}dz$$

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Defines:

$g_0(x)$: factors (locally), $g_1(z,x)$: factors (locally), $g_2(z,x)$: factor (locally) and $c$: center (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $C$: closed contour and $x$: real variable

Referenced by:

Table 18.10.1, (18.17.21_1)

Permalink:

http://dlmf.nist.gov/18.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.10(iii), §18.10 and Ch.18

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4: 14.24 Analytic Continuation

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14.24.1 $$P_{\nu }^{-\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={\mathrm{e}}^{s\nu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right)+\frac{2\mathrm{i}\sin \left(\left(\nu +\frac{1}{2}\right)s\pi \right){\mathrm{e}}^{-s\pi \mathrm{i}/2}}{\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\mu -\nu \right)}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.2 $${𝑸}_{\nu }^{\mu }\left(z{\mathrm{e}}^{s\pi \mathrm{i}}\right)={(-1)}^s{\mathrm{e}}^{-s\nu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right),$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.23, §14.24

Permalink:

http://dlmf.nist.gov/14.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.3 $$P_{\nu ,s}^{-\mu }\left(z\right)={\mathrm{e}}^{s\mu \pi \mathrm{i}}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Permalink:

http://dlmf.nist.gov/14.24.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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14.24.4 $${𝑸}_{\nu ,s}^{\mu }\left(z\right)={\mathrm{e}}^{-s\mu \pi \mathrm{i}}{𝑸}_{\nu }^{\mu }\left(z\right)-\frac{\pi \mathrm{i}\sin \left(s\mu \pi \right)}{\sin \left(\mu \pi \right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{P}_{\nu }^{-\mu }\left(z\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $z$: complex variable, $\mu$: general order, $\nu$: general degree and $s$: arbitrary integer

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.24.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.24 and Ch.14

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5: 18.30 Associated OP’s

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18.30.24 $$F_n(z)=p_n^{(0)}(z)/p_n(z)=\frac{A_0}{A_{0}z+B_0-}\frac{C_1}{A_{1}z+B_1-}\mathrm{\cdots }\frac{C_{n-1}}{A_{n-1}z+B_{n-1}}.$$

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Symbols:

$p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $A_n$: coefficient, $B_n$: coefficient and $C_n$: coefficient

Referenced by:

§18.30(vi), §18.30(vi)

Permalink:

http://dlmf.nist.gov/18.30.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

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18.30.25 $$\underset{n\to \mathrm{\infty }}{lim}{F}_n(x)=\underset{n\to \mathrm{\infty }}{lim}{p}_n^{(0)}(z)/p_n(z)=\frac{1}{{\mu }_0}{\int }_a^b\frac{d\mu (x)}{z-x},$$ $z\in ℂ\backslash [a,b]$.

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Chihara (1978, Chapter III, Theorem 4.3)(proved)

Referenced by:

§18.30(vi), §18.30(vii)

Permalink:

http://dlmf.nist.gov/18.30.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(vi), §18.30(vi), §18.30 and Ch.18

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6: 14.29 Generalizations

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14.29.1 $$\left(1-z^2\right)\frac{d^{2}w}{{dz}^2}-2z\frac{dw}{dz}+\left(\nu (\nu +1)-\frac{{\mu }_1^2}{2(1-z)}-\frac{{\mu }_2^2}{2(1+z)}\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.29

Permalink:

http://dlmf.nist.gov/14.29.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.29 and Ch.14

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7: 18.2 General Orthogonal Polynomials

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18.2.33 $$p_{n-1}^{(1)}(z)=\frac{1}{{\mu }_0}{\int }_a^b\frac{p_n(z)-p_n(x)}{z-x}d\mu (x),$$ $z\in ℂ\backslash [a,b]$, $n=1,2,\mathrm{\dots }$,

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Chihara (1978, Chapter III, §4)(proved)

Permalink:

http://dlmf.nist.gov/18.2.E33

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(x), §18.2 and Ch.18

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18.2.41 $$P_z(x,y)=\sum _{n=0}^{\mathrm{\infty }}\frac{p_n(x)p_n(y)}{h_n}{z}^n,$$ ,

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Symbols:

$(a,b)$: open interval, $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $z$: complex variable, $n$: nonnegative integer, $x$: real variable and $h_n$

Referenced by:

§18.18(vii), §18.2(xii)

Permalink:

http://dlmf.nist.gov/18.2.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §18.2(xii), §18.2(xii), §18.2 and Ch.18

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8: 14.25 Integral Representations

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14.25.1 $$P_{\nu }^{-\mu }\left(z\right)=\frac{{\left(z^2-1\right)}^{\mu /2}}{2^{\nu }\mathrm{\Gamma }\left(\mu -\nu \right)\mathrm{\Gamma }\left(\nu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\nu +1}}{{(z+\cosh t)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\mu >\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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14.25.2 $${𝑸}_{\nu }^{\mu }\left(z\right)=\frac{{\pi }^{1/2}{\left(z^2-1\right)}^{\mu /2}}{2^{\mu }\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu -\mu +1\right)}{\int }_0^{\mathrm{\infty }}\frac{{(\sinh t)}^{2\mu }}{{\left(z+{(z^2-1)}^{1/2}\cosh t\right)}^{\nu +\mu +1}}dt,$$ $\mathrm{\Re }\left(\nu +1\right)>\mathrm{\Re }\mu >-\frac{1}{2}$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.25

Permalink:

http://dlmf.nist.gov/14.25.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.25 and Ch.14

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9: 30.6 Functions of Complex Argument

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30.6.3 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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10: 30.11 Radial Spheroidal Wave Functions

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30.11.3 $$S_n^{m(j)}(z,\gamma )=\frac{{(1-z^{-2})}^{\frac{1}{2}m}}{A_n^{-m}({\gamma }^2)}\sum _{2k\ge m-n}{a}_{n,k}^{-m}({\gamma }^2){\psi }_{n+2k}^{(j)}(\gamma z).$$

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Defines:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function

Symbols:

$z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function, $A_n^m({\gamma }^2)$ , ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

A&S Ref:

21.9.1 (in different form)

Referenced by:

§30.11(i), §30.16(iii)

Permalink:

http://dlmf.nist.gov/30.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(i), §30.11 and Ch.30

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30.11.6

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, ${\psi }_k^{(j)}(z)$: spherical function and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iii), §30.11 and Ch.30

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30.11.7 $$𝒲\left\{S_n^{m(1)}(z,\gamma ),S_n^{m(2)}(z,\gamma )\right\}=\frac{1}{\gamma (z^2-1)}.$$

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Symbols:

$𝒲$: Wronskian, $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(iv), §30.11 and Ch.30

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30.11.8 $$S_n^{m(1)}(z,\gamma )=K_n^m(\gamma ){\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),$$

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Symbols:

$S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.1 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

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30.11.9 $$S_n^{m(2)}(z,\gamma )=\frac{(n-m)!}{(n+m)!}\frac{{(-1)}^{m+1}{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)}{\gamma K_n^m(\gamma )A_n^m({\gamma }^2)A_n^{-m}({\gamma }^2)},$$

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Symbols:

$!$: factorial (as in $n!$), $S_n^{m(j)}(z,\gamma )$: radial spheroidal wave function, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ , $K_n^m(\gamma )$: connection coefficient and ${\gamma }^2$: real parameter

A&S Ref:

21.10.2 (in different form)

Permalink:

http://dlmf.nist.gov/30.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.11(v), §30.11 and Ch.30

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