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21: 13.14 Definitions and Basic Properties

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13.14.20 $$M_{\kappa ,\mu }\left(z\right)\sim \mathrm{\Gamma }\left(1+2\mu \right){\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }/\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/13.14.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iv), §13.14 and Ch.13

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13.14.21 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa },$$ $\left|\mathrm{ph}z\right|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§33.21(i)

Permalink:

http://dlmf.nist.gov/13.14.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iv), §13.14 and Ch.13

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22: 13.2 Definitions and Basic Properties

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13.2.6 $$U(a,b,z)\sim z^{-a},$$ $z\to \mathrm{\infty }$, $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Defines:

$U(a,b,z)$: Kummer confluent hypergeometric function

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.2(i), §13.2(iv)

Permalink:

http://dlmf.nist.gov/13.2.E6

Encodings:

pMML, png, TeX

See also:

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

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13.2.23 $$𝐌(a,b,z)\sim {\mathrm{e}}^z{z}^{a-b}/\mathrm{\Gamma }\left(a\right),$$ $\left|\mathrm{ph}z\right|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/13.2.E23

Encodings:

pMML, png, TeX

See also:

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

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23: 18.25 Wilson Class: Definitions

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18.25.9 $$\sum _{y=0}^N{p}_n(y(y+\gamma +\delta +1))p_m(y(y+\gamma +\delta +1))\frac{\gamma +\delta +1+2y}{\gamma +\delta +1+y}{\omega }_y=h_n{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Referenced by:

§18.38(iii)

Permalink:

http://dlmf.nist.gov/18.25.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25 and Ch.18

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18.25.10 $$p_n(x)=R_n(x;\alpha ,\beta ,\gamma ,\delta ),$$ $\alpha +1=-N$,

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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18.25.11 $${\omega }_y=\frac{{\left(\alpha +1\right)}_y{\left(\beta +\delta +1\right)}_y{\left(\gamma +1\right)}_y{\left(\gamma +\delta +2\right)}_y}{{\left(-\alpha +\gamma +\delta +1\right)}_y{\left(-\beta +\gamma +1\right)}_y{\left(\delta +1\right)}_{y}y!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $y$: real variable and $\delta$: arbitrary small positive constant

Referenced by:

§18.25(i)

Permalink:

http://dlmf.nist.gov/18.25.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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18.25.13 $$p_n(x)=R_n(x;\gamma ,\delta ,N),$$

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Symbols:

$R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $N$: positive integer, $\delta$: arbitrary small positive constant, $p_n(x)$: polynomial of degree $n$ , $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.25(iii)

Permalink:

http://dlmf.nist.gov/18.25.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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18.25.14 $${\omega }_y=\frac{{(-1)}^y{\left(-N\right)}_y{\left(\gamma +1\right)}_y{\left(\gamma +\delta +1\right)}_2}{{\left(N+\gamma +\delta +2\right)}_y{\left(\delta +1\right)}_{y}y!},$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $y$: real variable, $N$: positive integer and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/18.25.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(iii), §18.25(iii), §18.25 and Ch.18

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24: 10.41 Asymptotic Expansions for Large Order

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10.41.12 $$I_{\nu }\left(\nu z\right)=\frac{{\mathrm{e}}^{\nu \eta }}{{(2\pi \nu )}^{\frac{1}{2}}{(1+z^2)}^{\frac{1}{4}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}\frac{U_k(p)}{{\nu }^k}+O\left(\frac{1}{z^{\mathrm{\ell }}}\right)\right),$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\eta$ , $p$ and $U_k(p)$: polynomial coefficient

Referenced by:

§10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E12

Encodings:

pMML, png, TeX

See also:

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

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10.41.13 $$K_{\nu }\left(\nu z\right)={\left(\frac{\pi }{2\nu }\right)}^{\frac{1}{2}}\frac{{\mathrm{e}}^{-\nu \eta }}{{(1+z^2)}^{\frac{1}{4}}}\left(\sum _{k=0}^{\mathrm{\ell }-1}{(-1)}^k\frac{U_k(p)}{{\nu }^k}+O\left(\frac{1}{z^{\mathrm{\ell }}}\right)\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\eta$ , $p$ and $U_k(p)$: polynomial coefficient

Referenced by:

§10.41(iv), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.41.E13

Encodings:

pMML, png, TeX

See also:

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

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25: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

${\left(a\right)}_n$: 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, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

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26: 11.6 Asymptotic Expansions

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11.6.1 $${𝐊}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.29

Referenced by:

§11.6(i), §11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.2 $${𝐌}_{\nu }\left(z\right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{\mathrm{\Gamma }\left(k+\frac{1}{2}\right){(\frac{1}{2}z)}^{\nu -2k-1}}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}-k\right)},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.6

Referenced by:

§11.6(i), §11.6(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.3 $${\int }_0^z{𝐊}_0\left(t\right)dt-\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k+1}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, ${𝐊}_{\nu }\left(z\right)$: Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.1.32

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.4 $${\int }_0^z{𝐌}_0\left(t\right)dt+\frac{2}{\pi }(\ln \left(2z\right)+\gamma )\sim \frac{2}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(2k)!(2k-1)!}{{(k!)}^2{(2z)}^{2k}},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\gamma$: Euler’s constant, ${𝐌}_{\nu }\left(z\right)$: modified Struve function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $k$: nonnegative integer and $\delta$: arbitrary small positive constant

A&S Ref:

12.2.8

Referenced by:

§11.6(i)

Permalink:

http://dlmf.nist.gov/11.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(i), §11.6 and Ch.11

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11.6.5 $${𝐇}_{\nu }\left(z\right),{𝐋}_{\nu }\left(z\right)\sim \frac{z}{\pi \nu \sqrt{2}}{\left(\frac{\mathrm{e}z}{2\nu }\right)}^{\nu },$$ $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐇}_{\nu }\left(z\right)$: Struve function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: real or complex order and $\delta$: arbitrary small positive constant

Referenced by:

(11.11.7)

Permalink:

http://dlmf.nist.gov/11.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §11.6(ii), §11.6 and Ch.11

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27: 7.1 Special Notation

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$x$	real variable.
$z$	complex variable.
…
$\delta$	arbitrary small positive constant.
…

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28: 12.1 Special Notation

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$x,y$	real variables.
$z$	complex variable.
…
$\delta$	arbitrary small positive constant.

►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

29: 10.7 Limiting Forms

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10.7.2 $$H_0^{(1)}\left(z\right)\sim -H_0^{(2)}\left(z\right)\sim (2\mathrm{i}/\pi )\ln z,$$

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Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

9.1.8

Referenced by:

§10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

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10.7.3 $$J_{\nu }\left(z\right)\sim {(\frac{1}{2}z)}^{\nu }/\mathrm{\Gamma }\left(\nu +1\right),$$ $\nu \ne -1,-2,-3,\mathrm{\dots }$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.7

Referenced by:

§10.7(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

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10.7.4 $$Y_{\nu }\left(z\right)\sim -(1/\pi )\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$$ $\mathrm{\Re }\nu >0$ or $\nu =-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.9

Referenced by:

§10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

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10.7.5 $$Y_{-\nu }\left(z\right)\sim -(1/\pi )\cos \left(\nu \pi \right)\mathrm{\Gamma }\left(\nu \right){(\frac{1}{2}z)}^{-\nu },$$ $\mathrm{\Re }\nu >0$, $\nu \ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

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10.7.6 $$Y_{\mathrm{i}\nu }\left(z\right)=\frac{\mathrm{i}\mathrm{csch}\left(\nu \pi \right)}{\mathrm{\Gamma }\left(1-\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{-\mathrm{i}\nu }-\frac{\mathrm{i}\coth \left(\nu \pi \right)}{\mathrm{\Gamma }\left(1+\mathrm{i}\nu \right)}{(\frac{1}{2}z)}^{\mathrm{i}\nu }+{\mathrm{e}}^{|\nu \mathrm{ph}z|}o\left(1\right),$$ $\nu \in ℝ$ and $\nu \ne 0$.

ⓘ

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, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\coth z$: hyperbolic cotangent function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than, $\mathrm{ph}$: phase, $ℝ$: real line, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.7(i), §10.7(i)

Permalink:

http://dlmf.nist.gov/10.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.7(i), §10.7 and Ch.10

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30: 12.9 Asymptotic Expansions for Large Variable

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12.9.1 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ ,

ⓘ

Symbols:

${\left(a\right)}_n$: 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 $n!$), $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.1 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

ⓘ

Symbols:

${\left(a\right)}_n$: 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 $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

►

12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma 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, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

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See also:

Annotations for §12.9(i), §12.9 and Ch.12

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