Olver hypergeometric function

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

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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.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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12: 13.2 Definitions and Basic Properties

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13.2.3 $$𝐌(a,b,z)=\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s}{\mathrm{\Gamma }\left(b+s\right)s!}{z}^s,$$

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Defines:

$𝐌(a,b,z)$: Olver’s confluent hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $s$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i)

Permalink:

http://dlmf.nist.gov/13.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.4 $$M(a,b,z)=\mathrm{\Gamma }\left(b\right)𝐌(a,b,z).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.2(ii)

Permalink:

http://dlmf.nist.gov/13.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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13.2.34 $$𝒲\left\{𝐌(a,b,z),U(a,b,z)\right\}=-z^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(a\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.22

Permalink:

http://dlmf.nist.gov/13.2.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vi), §13.2 and Ch.13

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13.2.35 $$𝒲\left\{𝐌(a,b,z),{\mathrm{e}}^{z}U(b-a,b,{\mathrm{e}}^{\pm \pi \mathrm{i}}z)\right\}={\mathrm{e}}^{\mp b\pi \mathrm{i}}{z}^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(b-a\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

13.1.23

Permalink:

http://dlmf.nist.gov/13.2.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vi), §13.2 and Ch.13

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13.2.36 $$𝒲\left\{z^{1-b}𝐌(a-b+1,2-b,z),U(a,b,z)\right\}=-z^{-b}{\mathrm{e}}^z/\mathrm{\Gamma }\left(a-b+1\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $𝒲$: Wronskian, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

A&S Ref:

13.1.24

Permalink:

http://dlmf.nist.gov/13.2.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(vi), §13.2 and Ch.13

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

ⓘ

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

ⓘ

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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14: 15.3 Graphics

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§15.3(i) Graphs

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Figure 15.3.2: $F(5,-10;1;x),-0.023\le x\le 1$. Magnify

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Figure 15.3.3: $F(1,-10;10;x),-3\le x\le 1$. Magnify

… ►In Figures 15.3.5 and 15.3.6, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 15.3.7: $|𝐅(-3,\frac{3}{5};u+\mathrm{i}v;\frac{1}{2})|,-6\le u\le 2,-2\le v\le 2$. Magnify 3D Help

15: 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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16: 14.19 Toroidal (or Ring) Functions

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14.19.2 $$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-\mu \right)}{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;1-{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$.

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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, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Referenced by:

§14.19(ii), Erratum (V1.0.1) for Equation (14.19.2)

Permalink:

http://dlmf.nist.gov/14.19.E2

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally the argument to $𝐅$ was incorrect and the condition on $\mu$ too weak:

$$P_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{\mathrm{\Gamma }\left(1-2\mu \right)2^{2\mu }}{\mathrm{\Gamma }\left(1-\mu \right){\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\frac{1}{2}-\mu ,\frac{1}{2}+\nu -\mu ;1-2\mu ;{\mathrm{e}}^{-2\xi }),$$ $\mu \ne \frac{1}{2}$.

Also, the factor multiplying $𝐅$ was rewritten to clarify the poles, which determine the correct condition on $\mu$.

Reported 2010-11-02 by Alvaro Valenzuela

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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14.19.3 $${𝑸}_{\nu -\frac{1}{2}}^{\mu }\left(\cosh \xi \right)=\frac{{\pi }^{1/2}{\left(1-{\mathrm{e}}^{-2\xi }\right)}^{\mu }}{{\mathrm{e}}^{(\nu +(1/2))\xi }}𝐅(\mu +\frac{1}{2},\nu +\mu +\frac{1}{2};\nu +1;{\mathrm{e}}^{-2\xi }).$$

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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, $\cosh z$: hyperbolic cosine function, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $\mu$: general order, $\nu$: general degree and $\xi >0$: variable

Permalink:

http://dlmf.nist.gov/14.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §14.19(ii), §14.19 and Ch.14

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17: 13.11 Series

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

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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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18: 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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19: 15.12 Asymptotic Approximations

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15.12.5 $$𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b-\lambda }{c};\frac{1}{2}-\frac{1}{2}z)=2^{(a+b-1)/2}\frac{{(z+1)}^{(c-a-b-1)/2}}{{(z-1)}^{c/2}}\sqrt{\zeta \sinh \zeta }{\left(\lambda +\frac{1}{2}a-\frac{1}{2}b\right)}^{1-c}\left(I_{c-1}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)(1+O({\lambda }^{-2}))+\frac{I_{c-2}\left((\lambda +\frac{1}{2}a-\frac{1}{2}b)\zeta \right)}{2\lambda +a-b}\left(\left(c-\frac{1}{2}\right)\left(c-\frac{3}{2}\right)\left(\frac{1}{\zeta }-\coth \zeta \right)+\frac{1}{2}(2c-a-b-1)(a+b-1)\tanh \left(\frac{1}{2}\zeta \right)+O({\lambda }^{-2})\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\tanh z$: hyperbolic tangent function, $I_{\nu }\left(z\right)$: 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, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/15.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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15.12.9 $${(z+1)}^{3\lambda /2}{(2\lambda )}^{c-1}𝐅(\genfrac{}{}{0pt}{}{a+\lambda ,b+2\lambda }{c};-z)={\lambda }^{-1/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(1/3))}\mathrm{Ai}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_0(\zeta )+O({\lambda }^{-1})\right)+{\lambda }^{-2/3}\left({\mathrm{e}}^{\pi \mathrm{i}(a-c+\lambda +(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{-2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)+{\mathrm{e}}^{\pi \mathrm{i}(c-a-\lambda -(2/3))}{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{\lambda }^{2/3}{\beta }^2\right)\right)\left(a_1(\zeta )+O({\lambda }^{-1})\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝐅(a,b;c;z)$ 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, $c$: real or complex parameter, $\zeta$: change of variable, $\beta$ and $a_j(\zeta )$: function

Referenced by:

§15.12(iii)

Permalink:

http://dlmf.nist.gov/15.12.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §15.12(iii), §15.12 and Ch.15

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20: 13.7 Asymptotic Expansions for Large Argument

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13.7.1 $$𝐌(a,b,x)\sim \frac{{\mathrm{e}}^x{x}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{x}^{-s},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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13.7.2 $$𝐌(a,b,z)\sim \frac{{\mathrm{e}}^z{z}^{a-b}}{\mathrm{\Gamma }\left(a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_s{\left(b-a\right)}_s}{s!}{z}^{-s}+\frac{{\mathrm{e}}^{\pm \pi \mathrm{i}a}{z}^{-a}}{\mathrm{\Gamma }\left(b-a\right)}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.1 (with corrected region of validity)

Referenced by:

§13.2(iv), §13.7(ii), §13.9(i)

Permalink:

http://dlmf.nist.gov/13.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.7(i), §13.7 and Ch.13

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