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21: 14.5 Special Values

§14.5 Special Values

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14.5.5 $${𝖯}_0\left(x\right)=P_0\left(x\right)=1,$$

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Symbols:

${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable

A&S Ref:

8.4.1

Permalink:

http://dlmf.nist.gov/14.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(ii), §14.5 and Ch.14

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14.5.6 $${𝖯}_1\left(x\right)=P_1\left(x\right)=x.$$

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Symbols:

${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind and $x$: real variable

A&S Ref:

8.4.3

Permalink:

http://dlmf.nist.gov/14.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §14.5(ii), §14.5 and Ch.14

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§14.5(v) $\mu =0$, $\nu =\pm \frac{1}{2}$

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22: 14.2 Differential Equations

… ►Standard solutions: ${𝖯}_{\nu }\left(\pm x\right)$, ${𝖰}_{\nu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}\left(\pm x\right)$, $P_{\nu }\left(\pm x\right)$, $Q_{\nu }\left(\pm x\right)$, $Q_{-\nu -1}\left(\pm x\right)$. … ►

§14.2(ii) Associated Legendre Equation

… ►Standard solutions: ${𝖯}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$, ${𝖰}_{\nu }^{\mu }\left(\pm x\right)$, ${𝖰}_{-\nu -1}^{\mu }\left(\pm x\right)$, $P_{\nu }^{\mu }\left(\pm x\right)$, $P_{\nu }^{-\mu }\left(\pm x\right)$, ${𝑸}_{\nu }^{\mu }\left(\pm x\right)$, ${𝑸}_{-\nu -1}^{\mu }\left(\pm x\right)$. … ►Ferrers functions and the associated Legendre functions are related to the Legendre functions by the equations ${𝖯}_{\nu }^0\left(x\right)={𝖯}_{\nu }\left(x\right)$, ${𝖰}_{\nu }^0\left(x\right)={𝖰}_{\nu }\left(x\right)$, $P_{\nu }^0\left(x\right)=P_{\nu }\left(x\right)$, $Q_{\nu }^0\left(x\right)=Q_{\nu }\left(x\right)$, ${𝑸}_{\nu }^0\left(x\right)={𝑸}_{\nu }\left(x\right)=Q_{\nu }\left(x\right)/\mathrm{\Gamma }\left(\nu +1\right)$. … ►

§14.2(iv) Wronskians and Cross-Products

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23: 14.6 Integer Order

§14.6 Integer Order

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14.6.1 $${𝖯}_{\nu }^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m{𝖯}_{\nu }\left(x\right)}{{dx}^m},$$

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Symbols:

: Ferrers function of the first kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.6

Permalink:

http://dlmf.nist.gov/14.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

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14.6.2 $${𝖰}_{\nu }^m\left(x\right)={(-1)}^m{\left(1-x^2\right)}^{m/2}\frac{d^m{𝖰}_{\nu }\left(x\right)}{{dx}^m}.$$

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Symbols:

: Ferrers function of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

A&S Ref:

8.6.7

Permalink:

http://dlmf.nist.gov/14.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.6(i), §14.6 and Ch.14

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14.6.6 $${𝖯}_{\nu }^{-m}\left(x\right)={\left(1-x^2\right)}^{-m/2}{\int }_x^1\mathrm{\dots }{\int }_x^1{𝖯}_{\nu }\left(x\right){\left(dx\right)}^m.$$

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Symbols:

: Ferrers function of the first kind, $dx$: differential of $x$ , $\int$: integral, ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$: Legendre function of the first kind, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.12(i), Erratum (V1.0.28) for Equation (14.6.6)

Permalink:

http://dlmf.nist.gov/14.6.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

The right-hand side has been corrected by replacing the Legendre function $P_{\nu }\left(x\right)$ with the Ferrers function ${𝖯}_{\nu }\left(x\right)$.

See also:

Annotations for §14.6(ii), §14.6 and Ch.14

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24: 14.34 Software

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25: Tom M. Apostol

… ►He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors. …

26: 14.20 Conical (or Mehler) Functions

… ►Another real-valued solution ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ of (14.20.1) was introduced in Dunster (1991). …${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists except when $\mu =\frac{1}{2},\frac{3}{2},\mathrm{\dots }$ and $\tau =0$; compare §14.3(i). …provided that ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ exists. … ►Approximations for ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{\mu }\left(x\right)$ and ${\widehat{𝖰}}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$ can then be achieved via (14.9.7) and (14.20.3). … ►For zeros of ${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }\left(x\right)$ see Hobson (1931, §237). …

27: 14.13 Trigonometric Expansions

§14.13 Trigonometric Expansions

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14.13.1 $${𝖯}_{\nu }^{\mu }\left(\cos \theta \right)=\frac{2^{\mu +1}{(\sin \theta )}^{\mu }}{{\pi }^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\sin \left((\nu +\mu +2k+1)\theta \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Volkmer (2021)

A&S Ref:

8.7.1

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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14.13.2 $${𝖰}_{\nu }^{\mu }\left(\cos \theta \right)={\pi }^{1/2}{2}^{\mu }{(\sin \theta )}^{\mu }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\nu +\mu +k+1\right)}{\mathrm{\Gamma }\left(\nu +k+\frac{3}{2}\right)}\frac{{\left(\mu +\frac{1}{2}\right)}_k}{k!}\cos \left((\nu +\mu +2k+1)\theta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $\sin z$: sine function, $\mu$: general order, $\nu$: general degree and : variable

Source:

Cohl et al. (2021, Theorem 6.2)

A&S Ref:

8.7.2

Referenced by:

§14.13, Erratum (V1.1.3) for Equations (14.13.1), (14.13.2)

Permalink:

http://dlmf.nist.gov/14.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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14.13.3 $${𝖯}_n\left(\cos \theta \right)=\frac{2^{2n+2}{(n!)}^2}{\pi (2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\sin \left((n+2k+1)\theta \right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$: Ferrers function of the first kind, $\sin z$: sine function, $n$: nonnegative integer and : variable

A&S Ref:

8.7.3

Permalink:

http://dlmf.nist.gov/14.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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14.13.4 $${𝖰}_n\left(\cos \theta \right)=\frac{2^{2n+1}{(n!)}^2}{(2n+1)!}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\mathrm{\cdots }(2k-1)}{k!}\frac{(n+1)(n+2)\mathrm{\cdots }(n+k)}{(2n+3)(2n+5)\mathrm{\cdots }(2n+2k+1)}\cos \left((n+2k+1)\theta \right),$$

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Symbols:

$\cos z$: cosine function, $!$: factorial (as in $n!$), ${𝖰}_{\nu }\left(x\right)={𝖰}_{\nu }^0\left(x\right)$: Ferrers function of the second kind, $n$: nonnegative integer and : variable

A&S Ref:

8.7.4

Permalink:

http://dlmf.nist.gov/14.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §14.13 and Ch.14

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28: 14.23 Values on the Cut

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14.23.1 $$P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)={\mathrm{e}}^{\mp \mu \pi \mathrm{i}/2}{𝖯}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.2 $${𝑸}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)=\frac{{\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}\left({𝖰}_{\nu }^{\mu }\left(x\right)\mp \frac{1}{2}\pi \mathrm{i}{𝖯}_{\nu }^{\mu }\left(x\right)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.23

Permalink:

http://dlmf.nist.gov/14.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.4 $${𝖯}_{\nu }^{\mu }\left(x\right)={\mathrm{e}}^{\pm \mu \pi \mathrm{i}/2}{P}_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right),$$

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Symbols:

: Ferrers function of the first kind, $P_{\nu }^{\mu }\left(z\right)$: 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, $x$: 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:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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14.23.5 $${𝖰}_{\nu }^{\mu }\left(x\right)=\frac{1}{2}\mathrm{\Gamma }\left(\nu +\mu +1\right)\left({\mathrm{e}}^{-\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x+\mathrm{i}0\right)+{\mathrm{e}}^{\mu \pi \mathrm{i}/2}{𝑸}_{\nu }^{\mu }\left(x-\mathrm{i}0\right)\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the second 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, $x$: 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:

pMML, png, TeX

See also:

Annotations for §14.23 and Ch.14

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

29: 30.8 Expansions in Series of Ferrers Functions

§30.8 Expansions in Series of Ferrers Functions

… ►where ${𝖯}_{n+2k}^m\left(x\right)$ is the Ferrers function of the first kind (§14.3(i)), $R=\lfloor \frac{1}{2}(n-m)\rfloor$, and the coefficients $a_{n,k}^m({\gamma }^2)$ are given by … ►

30.8.9 $${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)=\sum _{k=-\mathrm{\infty }}^{-N-1}{(-1)}^{k}a_{}^{\prime }{}_{n,k}{}^m({\gamma }^2){𝖯}_{n+2k}^m\left(x\right)+\sum _{k=-N}^{\mathrm{\infty }}{(-1)}^k{a}_{n,k}^m({\gamma }^2){𝖰}_{n+2k}^m\left(x\right),$$

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Symbols:

: Ferrers function of the first kind, : Ferrers function of the second 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, ${\gamma }^2$: real parameter, $a_{n,k}^m({\gamma }^2)$: coefficients and $N$: upper limit

A&S Ref:

21.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/30.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §30.8(ii), §30.8 and Ch.30

►where ${𝖯}_n^m$ and ${𝖰}_n^m$ are again the Ferrers functions and $N=\lfloor \frac{1}{2}(n+m)\rfloor$. …

30: 14.32 Methods of Computation

§14.32 Methods of Computation

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