Olver’s

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21: 14.25 Integral Representations

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

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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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22: 8.5 Confluent Hypergeometric Representations

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8.5.2 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-z}𝐌(1,1+a,z)=𝐌(a,1+a,-z).$$

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Symbols:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $a$: parameter

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.5 and Ch.8

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23: 14.22 Graphics

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24: 14.9 Connection Formulas

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14.9.12 $$\cos \left(\nu \pi \right)P_{\nu }^{-\mu }\left(x\right)=-\frac{{𝑸}_{\nu }^{\mu }\left(x\right)}{\mathrm{\Gamma }\left(\mu -\nu \right)}+\frac{{𝑸}_{-\nu -1}^{\mu }\left(x\right)}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}.$$

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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, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.2 (modified)

Referenced by:

§14.20(i), §14.8(ii), §14.8(iii)

Permalink:

http://dlmf.nist.gov/14.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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14.9.14 $${𝑸}_{\nu }^{-\mu }\left(x\right)={𝑸}_{\nu }^{\mu }\left(x\right),$$

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Symbols:

${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.6 (modified)

Referenced by:

§14.23, §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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14.9.15 $$\frac{2\sin \left(\mu \pi \right)}{\pi }{𝑸}_{\nu }^{\mu }\left(x\right)=\frac{P_{\nu }^{\mu }\left(x\right)}{\mathrm{\Gamma }\left(\nu +\mu +1\right)}-\frac{P_{\nu }^{-\mu }\left(x\right)}{\mathrm{\Gamma }\left(\nu -\mu +1\right)}.$$

ⓘ

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, $\sin z$: sine function, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.5 (modified)

Permalink:

http://dlmf.nist.gov/14.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iii), §14.9 and Ch.14

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14.9.16 $${𝑸}_{\nu }^{\mu }\left(x\right)={\left(\frac{1}{2}\pi \right)}^{1/2}{\left(x^2-1\right)}^{-1/4}{P}_{-\mu -(1/2)}^{-\nu -(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

A&S Ref:

8.2.7 (modified)

Referenced by:

§14.19(v), §14.21(iii), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

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14.9.17 $$P_{\nu }^{\mu }\left(x\right)={(2/\pi )}^{1/2}{\left(x^2-1\right)}^{-1/4}{𝑸}_{-\mu -(1/2)}^{\nu +(1/2)}\left(x{\left(x^2-1\right)}^{-1/2}\right).$$

ⓘ

Symbols:

$P_{\nu }^{\mu }\left(z\right)$: 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, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.19(v), §14.9(iv)

Permalink:

http://dlmf.nist.gov/14.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §14.9(iv), §14.9 and Ch.14

25: 14.8 Behavior at Singularities

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14.8.9 $${𝑸}_{\nu }\left(x\right)=-\frac{\ln \left(x-1\right)}{2\mathrm{\Gamma }\left(\nu +1\right)}+\frac{\frac{1}{2}\ln 2-\gamma -\psi \left(\nu +1\right)}{\mathrm{\Gamma }\left(\nu +1\right)}+O\left(\left(x-1\right)\ln \left(x-1\right)\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $\nu$: general degree

Referenced by:

Erratum (V1.1.5) for Equation (14.8.9)

Permalink:

http://dlmf.nist.gov/14.8.E9

Encodings:

pMML, png, TeX

Correction (effective with 1.1.5):

The symbol $O\left(x-1\right)$ has been corrected to be $O\left(\left(x-1\right)\ln \left(x-1\right)\right)$.

Suggested 2022-02-09 by Mark Ashbaugh

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

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14.8.10 $${𝑸}_{-n}\left(x\right)\to {(-1)}^{n+1}(n-1)!,$$ $n=1,2,3,\mathrm{\dots }$,

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Symbols:

$!$: factorial (as in $n!$), ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable and $n$: nonnegative integer

Referenced by:

§14.8(ii)

Permalink:

http://dlmf.nist.gov/14.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

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14.8.11 $${𝑸}_{\nu }^{\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(\mu \right)}{2\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\left(\frac{2}{x-1}\right)}^{\mu /2},$$ $\mathrm{\Re }\mu >0$, $\nu +\mu \ne -1,-2,-3,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\mathrm{\Re }$: real part, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.24

Permalink:

http://dlmf.nist.gov/14.8.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(ii), §14.8 and Ch.14

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14.8.15 $${𝑸}_{\nu }^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}}{\mathrm{\Gamma }\left(\nu +\frac{3}{2}\right){(2x)}^{\nu +1}},$$ $\nu \ne -\frac{3}{2},-\frac{5}{2},-\frac{7}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.8(iii)

Permalink:

http://dlmf.nist.gov/14.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(iii), §14.8 and Ch.14

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14.8.16 $${𝑸}_{-n-(1/2)}^{\mu }\left(x\right)\sim \frac{{\pi }^{1/2}\mathrm{\Gamma }\left(\mu +n+\frac{1}{2}\right)}{n!\mathrm{\Gamma }\left(\mu -n+\frac{1}{2}\right){(2x)}^{n+(1/2)}},$$ $n=1,2,3,\mathrm{\dots }$, $\mu -n+\frac{1}{2}\ne 0,-1,-2,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $x$: real variable, $n$: nonnegative integer and $\mu$: general order

Referenced by:

§14.21(iii), §14.8(iii)

Permalink:

http://dlmf.nist.gov/14.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §14.8(iii), §14.8 and Ch.14

26: 14.5 Special Values

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14.5.9 $${𝑸}_0\left(x\right)=\frac{1}{2}\ln \left(\frac{x+1}{x-1}\right),$$

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Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

A&S Ref:

8.4.2 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E9

Encodings:

pMML, png, TeX

See also:

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

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14.5.10 $${𝑸}_1\left(x\right)=\frac{x}{2}\ln \left(\frac{x+1}{x-1}\right)-1.$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

A&S Ref:

8.4.4 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E10

Encodings:

pMML, png, TeX

See also:

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

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14.5.26 $${𝑸}_{\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}\cosh \xi \mathrm{sech}\left(\frac{1}{2}\xi \right)K\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right)-4{\pi }^{-1/2}\cosh \left(\frac{1}{2}\xi \right)E\left(\mathrm{sech}\left(\frac{1}{2}\xi \right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.7 (in slightly different form)

Permalink:

http://dlmf.nist.gov/14.5.E26

Encodings:

pMML, png, TeX

See also:

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

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14.5.27 $${𝑸}_{-\frac{1}{2}}\left(\cosh \xi \right)=2{\pi }^{-1/2}{\mathrm{e}}^{-\xi /2}K\left({\mathrm{e}}^{-\xi }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $\xi >0$: variable

A&S Ref:

8.13.4 (modified)

Permalink:

http://dlmf.nist.gov/14.5.E27

Encodings:

pMML, png, TeX

See also:

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

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14.5.30 $${𝑸}_2\left(x\right)=\frac{3x^2-1}{8}\ln \left(\frac{x+1}{x-1}\right)-\frac{3}{4}x.$$

ⓘ

Symbols:

$\ln z$: principal branch of logarithm function, ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function and $x$: real variable

Referenced by:

(14.5.30)

Permalink:

http://dlmf.nist.gov/14.5.E30

Encodings:

pMML, png, TeX

Addition (effective with 1.0.7):

(14.5.30) has been added to this section.

See also:

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

27: 2 Asymptotic Approximations

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28: 14.6 Integer Order

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14.6.5 $${\left(\nu +1\right)}_m{𝑸}_{\nu }^m\left(x\right)={(-1)}^m{\left(x^2-1\right)}^{m/2}\frac{d^m{𝑸}_{\nu }\left(x\right)}{{dx}^m}.$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝑸}_{\nu }^{\mu }\left(z\right)$: Olver’s associated Legendre function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , ${𝑸}_{\nu }\left(z\right)={𝑸}_{\nu }^0\left(z\right)$: Olver’s associated Legendre function, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Referenced by:

§14.21(iii), §14.6(i)

Permalink:

http://dlmf.nist.gov/14.6.E5

Encodings:

pMML, png, TeX

See also:

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

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29: 15.9 Relations to Other Functions

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15.9.17 $$𝐅(\genfrac{}{}{0pt}{}{a,a+\frac{1}{2}}{c};z)=2^{c-1}{z}^{(1-c)/2}{(1-z)}^{-a+((c-1)/2)}{P}_{2a-c}^{1-c}\left(\frac{1}{\sqrt{1-z}}\right),$$ and .

ⓘ

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, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.18 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b+\frac{1}{2}};z)=2^{a+b-(1/2)}{(-z)}^{(-a-b+(1/2))/2}{P}_{a-b-(1/2)}^{-a-b+(1/2)}\left(\sqrt{1-z}\right),$$ .

ⓘ

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, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.19 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{a-b+1};z)=z^{(b-a)/2}{(1-z)}^{-b}{P}_{-b}^{b-a}\left(\frac{1+z}{1-z}\right),$$ and .

ⓘ

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, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.20 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{\frac{1}{2}(a+b+1)};z)={\left(-z(1-z)\right)}^{(1-a-b)/4}{P}_{(a-b-1)/2}^{(1-a-b)/2}\left(1-2z\right),$$ .

ⓘ

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, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $b$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.9.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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15.9.21 $$𝐅(\genfrac{}{}{0pt}{}{a,1-a}{c};z)={\left(\frac{-z}{1-z}\right)}^{(1-c)/2}{P}_{-a}^{1-c}\left(1-2z\right),$$ .

ⓘ

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, $\mathrm{ph}$: phase, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.9(iv)

Permalink:

http://dlmf.nist.gov/15.9.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §15.9(iv), §15.9 and Ch.15

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30: 13.11 Series

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13.11.3 $$𝐌(a,b,z)={\mathrm{e}}^{\frac{1}{2}z}\sum _{s=0}^{\mathrm{\infty }}{A}_s{\left(b-2a\right)}^{\frac{1}{2}(1-b-s)}{\left(\frac{1}{2}z\right)}^{\frac{1}{2}(1-b+s)}{J}_{b-1+s}\left(\sqrt{2z(b-2a)}\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $s$: nonnegative integer, $z$: complex variable and $A_n$: coefficients

Source:

Slater (1960, §3.8)

A&S Ref:

13.3.7

Referenced by:

§13.11, Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/13.11.E3

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §13.11 and Ch.13

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