scaling of variables and parameters

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11: 15.8 Transformations of Variable

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15.8.1 $$𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)={(1-z)}^{-a}𝐅(\genfrac{}{}{0pt}{}{a,c-b}{c};\frac{z}{z-1})={(1-z)}^{-b}𝐅(\genfrac{}{}{0pt}{}{c-a,b}{c};\frac{z}{z-1})={(1-z)}^{c-a-b}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c};z),$$ .

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Symbols:

$\pi$: 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, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.10(ii), §15.13, §15.16, (15.4.27), §15.4(i), §15.4(iii), §15.8(i), §15.8(ii), §15.8(iii), §8.17(i)

Permalink:

http://dlmf.nist.gov/15.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

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15.8.2 $$\frac{\sin \left(\pi (b-a)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{{(-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}𝐅(\genfrac{}{}{0pt}{}{a,a-c+1}{a-b+1};\frac{1}{z})-\frac{{(-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{b,b-c+1}{b-a+1};\frac{1}{z}),$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.12(i), §15.16, §15.8(i), §15.8(i), §15.8(ii), §15.8(ii), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

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15.8.3 $$\frac{\sin \left(\pi (b-a)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{{(1-z)}^{-a}}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-a\right)}𝐅(\genfrac{}{}{0pt}{}{a,c-b}{a-b+1};\frac{1}{1-z})-\frac{{(1-z)}^{-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{b,c-a}{b-a+1};\frac{1}{1-z}),$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.8(i), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

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15.8.4 $$\frac{\sin \left(\pi (c-a-b)\right)}{\pi }𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)=\frac{1}{\mathrm{\Gamma }\left(c-a\right)\mathrm{\Gamma }\left(c-b\right)}𝐅(\genfrac{}{}{0pt}{}{a,b}{a+b-c+1};1-z)-\frac{{(1-z)}^{c-a-b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)}𝐅(\genfrac{}{}{0pt}{}{c-a,c-b}{c-a-b+1};1-z),$$ , .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\sin z$: sine function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Referenced by:

§15.16, §15.19(iv), §15.2(i), §15.4(ii), §15.8(i), §15.8(i), §15.8(ii)

Permalink:

http://dlmf.nist.gov/15.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(i), §15.8 and Ch.15

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15.8.12 $$𝐅(a,b;a+b-m;z)={(1-z)}^{-m}𝐅(\tilde{a},\tilde{b};\tilde{a}+\tilde{b}+m;z),$$ $\tilde{a}=a-m,\tilde{b}=b-m$.

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Symbols:

$𝐅(a,b;c;z)$ 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, $b$: real or complex parameter and $m$: integer

Permalink:

http://dlmf.nist.gov/15.8.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(ii), §15.8 and Ch.15

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12: 36.6 Scaling Relations

§36.6 Scaling Relations

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Diffraction Catastrophe Scaling

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Indices for $k$-Scaling of Magnitude of ${\mathrm{\Psi }}_K$ or ${\mathrm{\Psi }}^{(\mathrm{U})}$ (Singularity Index)

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Indices for $k$-Scaling of Coordinates $x_m$

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Indices for $k$-Scaling of $𝐱$ Hypervolume

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13: 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 .

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

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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, $\mathrm{ph}$: 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 .

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

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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, $\mathrm{ph}$: 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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14: 10.39 Relations to Other Functions

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10.39.10 $$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;z^2/(4\lambda \mu )),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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15: 16.2 Definition and Analytic Properties

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16.2.5 $${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)/\left(\mathrm{\Gamma }\left(b_1\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q\right)\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{\mathrm{\Gamma }\left(b_1+k\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q+k\right)}\frac{z^k}{k!};$$

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Defines:

${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.2(v)

Permalink:

http://dlmf.nist.gov/16.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(v), §16.2 and Ch.16

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16: 21.9 Integrable Equations

… ►Here $x$ and $y$ are spatial variables, $t$ is time, and $u(x,y,t)$ is the elevation of the surface wave. All quantities are made dimensionless by a suitable scaling transformation. …These parameters, including $𝛀$, are not free: they are determined by a compact, connected Riemann surface (Krichever (1976)), or alternatively by an appropriate initial condition $u(x,y,0)$ (Deconinck and Segur (1998)). …

17: 10.16 Relations to Other Functions

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10.16.10 $$J_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;-z^2/(4\lambda \mu )),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.70 (modified)

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.16, §10.16 and Ch.10

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18: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

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§13.8(iii) Large $a$

… ► … ►where ${\mathrm{\Gamma }}^{\ast }\left(a\right)$ is the scaled gamma function defined in (5.11.3). …

19: 10.22 Integrals

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10.22.49 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)dt=\frac{{(\frac{1}{2}b)}^{\nu }}{a^{\mu +\nu }}\mathrm{\Gamma }\left(\mu +\nu \right)𝐅(\frac{\mu +\nu }{2},\frac{\mu +\nu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\mu +\nu \right)>0,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Laplace transform, Mellin transform

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.50 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{Y}_{\nu }\left(bt\right)dt=\cot \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{\nu }\mathrm{\Gamma }\left(\mu +\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu +\nu )}}𝐅(\frac{\mu +\nu }{2},\frac{1-\mu +\nu }{2};\nu +1;\frac{b^2}{a^2+b^2})-\csc \left(\nu \pi \right)\frac{{(\frac{1}{2}b)}^{-\nu }\mathrm{\Gamma }\left(\mu -\nu \right)}{{(a^2+b^2)}^{\frac{1}{2}(\mu -\nu )}}𝐅(\frac{\mu -\nu }{2},\frac{1-\mu -\nu }{2};1-\nu ;\frac{b^2}{a^2+b^2}),$$ $\mathrm{\Re }\mu >|\mathrm{\Re }\nu |,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$.

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\cot z$: cotangent function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.22.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

►For the hypergeometric function $𝐅$ see §15.2(i). … ►

10.22.56 $${\int }_0^{\mathrm{\infty }}\frac{J_{\mu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{a^{\mu }\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{b}^{\mu -\lambda +1}\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{1}{2}(\mu +\nu -\lambda +1),\frac{1}{2}(\mu -\nu -\lambda +1);\mu +1;\frac{a^2}{b^2}),$$ , $\mathrm{\Re }\left(\mu +\nu +1\right)>\mathrm{\Re }\lambda >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

A&S Ref:

11.4.33, 11.4.34

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E56

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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10.22.58 $${\int }_0^{\mathrm{\infty }}\frac{J_{\nu }\left(at\right)J_{\nu }\left(bt\right)}{t^{\lambda }}dt=\frac{{(ab)}^{\nu }\mathrm{\Gamma }\left(\nu -\frac{1}{2}\lambda +\frac{1}{2}\right)}{2^{\lambda }{(a^2+b^2)}^{\nu -\frac{1}{2}\lambda +\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}\lambda +\frac{1}{2}\right)}𝐅(\frac{2\nu +1-\lambda }{4},\frac{2\nu +3-\lambda }{4};\nu +1;\frac{4a^2{b}^2}{{(a^2+b^2)}^2}),$$ $a\ne b$, $\mathrm{\Re }\left(2\nu +1\right)>\mathrm{\Re }\lambda >-1$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function and $\nu$: complex parameter

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/10.22.E58

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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20: 10.41 Asymptotic Expansions for Large Order

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10.41.1 $$I_{\nu }\left(z\right)\sim \frac{1}{\sqrt{2\pi \nu }}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.41.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(i), §10.41 and Ch.10

►

10.41.2 $$K_{\nu }\left(z\right)\sim \sqrt{\frac{\pi }{2\nu }}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{-\nu }.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.41.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.41(i), §10.41 and Ch.10

►

§10.41(ii) Uniform Expansions for Real Variable

… ►

§10.41(iii) Uniform Expansions for Complex Variable

… ►This is because $A_k(\zeta )$ and ${\zeta }^{-\frac{1}{2}}{B}_k(\zeta ),k=0,1,\mathrm{\dots }$, do not form an asymptotic scale (§2.1(v)) as $\zeta \to +\mathrm{\infty }$; see Olver (1997b, pp. 422–425). …