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11: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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12: 10.52 Limiting Forms

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10.52.5 $${𝗂}_n^{(1)}\left(z\right)\sim {𝗂}_n^{(2)}\left(z\right)\sim \frac{1}{2}{z}^{-1}{\mathrm{e}}^z,$$ ,

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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, $\mathrm{ph}$: phase, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, ${𝗂}_n^{(2)}\left(z\right)$: modified spherical Bessel function, $n$: integer, $z$: complex variable and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.52.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.52(ii), §10.52 and Ch.10

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13: 18.26 Wilson Class: Continued

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18.26.4_1 $$R_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)=Q_y(n;\gamma ,\delta ,N),$$

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Symbols:

$Q_n(x;\alpha ,\beta ,N)$: Hahn polynomial, $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

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18.26.4_2 $$R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )=R_y(n(n+\alpha +\beta +1);\gamma ,\delta ,\alpha ,\beta ).$$

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Symbols:

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

Referenced by:

§18.26(i), Erratum (V1.2.0) Section 18.26

Permalink:

http://dlmf.nist.gov/18.26.E4_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.26(i), §18.26(i), §18.26 and Ch.18

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18.26.16 $$\frac{{\mathrm{\Delta }}_y\left(R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta )\right)}{{\mathrm{\Delta }}_y\left(y(y+\gamma +\delta +1)\right)}=\frac{n(n+\alpha +\beta +1)}{(\alpha +1)(\beta +\delta +1)(\gamma +1)}{R}_{n-1}(y(y+\gamma +\delta +2);\alpha +1,\beta +1,\gamma +1,\delta ).$$

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Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $\mathrm{\Delta }$: forward difference operator, $y$: real variable, $\delta$: arbitrary small positive constant and $n$: nonnegative integer

Referenced by:

§16.4(iii)

Permalink:

http://dlmf.nist.gov/18.26.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iii), §18.26 and Ch.18

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18.26.20 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-y,-y+\beta -\gamma }{\beta +\delta +1};z){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{y-N,y+\gamma +1}{-\delta -N};z)=\sum _{n=0}^N\frac{{\left(-N\right)}_n{\left(\gamma +1\right)}_n}{{\left(-\delta -N\right)}_{n}n!}{R}_n(y(y+\gamma +\delta +1);-N-1,\beta ,\gamma ,\delta )z^n.$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $!$: factorial (as in $n!$), $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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18.26.21 $${(1-z)}^y{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{y-N,y+\gamma +1}{-\delta -N};z)=\sum _{n=0}^N\frac{{\left(\gamma +1\right)}_n{\left(-N\right)}_n}{{\left(-\delta -N\right)}_{n}n!}{R}_n(y(y+\gamma +\delta +1);\gamma ,\delta ,N)z^n.$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $R_n(x;\gamma ,\delta ,N)$: dual Hahn polynomial, $!$: factorial (as in $n!$), $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§18.26(iv)

Permalink:

http://dlmf.nist.gov/18.26.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.26(iv), §18.26(iv), §18.26 and Ch.18

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14: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

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§33.5(iii) Small $|\eta |$

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15: 8.11 Asymptotic Approximations and Expansions

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8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.5 $$P(a,z)\sim \frac{z^a{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1+a\right)}\sim {(2\pi a)}^{-\frac{1}{2}}{\mathrm{e}}^{a-z}{(z/a)}^a,$$ $a\to \mathrm{\infty }$, $|\mathrm{ph}a|\le \pi -\delta$.

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Symbols:

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

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(ii), §8.11 and Ch.8

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8.11.6 $$\gamma (a,z)\sim -z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ , $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Notes:

See Paris (2002b)

Referenced by:

§8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, . It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where and $|\mathrm{ph}a|\le \frac{\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

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8.11.7 $$\mathrm{\Gamma }(a,z)\sim z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-a)}^k{b}_k(\lambda )}{{(z-a)}^{2k+1}},$$ $\lambda >1$, $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $x$: real variable, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $\delta$: small positive constant, $\lambda$: parameter and $b_k(\lambda )$: coefficients

Referenced by:

§8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E7

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

Originally this equation was stated for real variables $a$ and $x(=\lambda a)$, $\lambda >1$. It has been extended to allow for complex variables $a$ and $z(=\lambda a)$, where $\lambda >1$ and $|\mathrm{ph}a|\le \frac{3\pi }{2}-\delta$.

See also:

Annotations for §8.11(iii), §8.11 and Ch.8

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8.11.12 $$\mathrm{\Gamma }(z,z)\sim z^{z-1}{\mathrm{e}}^{-z}\left(\sqrt{\frac{\pi }{2}}{z}^{\frac{1}{2}}-\frac{1}{3}+\frac{\sqrt{2\pi }}{24z^{\frac{1}{2}}}-\frac{4}{135z}+\frac{\sqrt{2\pi }}{576z^{\frac{3}{2}}}+\frac{8}{2835z^2}+\mathrm{\dots }\right),$$ $|\mathrm{ph}z|\le 2\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{ph}$: phase, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

6.5.35 (Modified form)

Referenced by:

§8.11(v), Firefox 3 users should upgrade, Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)

Permalink:

http://dlmf.nist.gov/8.11.E12

Encodings:

pMML, png, TeX

Addition (effective with 1.0.14):

The sector of validity for the above equation was increased from $|\mathrm{ph}z|\le \pi -\delta$ to $|\mathrm{ph}z|\le 2\pi -\delta$.

See also:

Annotations for §8.11(v), §8.11 and Ch.8

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16: 10.17 Asymptotic Expansions for Large Argument

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10.17.3 $$J_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}-\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.5

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.49(i), §10.61(v), §10.7(ii)

Permalink:

http://dlmf.nist.gov/10.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.4 $$Y_{\nu }\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.6

Referenced by:

§10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.5 $$H_{\nu }^{(1)}\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{\mathrm{i}\omega }\sum _{k=0}^{\mathrm{\infty }}{\mathrm{i}}^k\frac{a_k(\nu )}{z^k},$$ $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$,

ⓘ

Symbols:

$H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.7

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.41(v)

Permalink:

http://dlmf.nist.gov/10.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.6 $$H_{\nu }^{(2)}\left(z\right)\sim {\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-\mathrm{i}\omega }\sum _{k=0}^{\mathrm{\infty }}{(-\mathrm{i})}^k\frac{a_k(\nu )}{z^k},$$ $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$,

ⓘ

Symbols:

$H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $a_k(\nu )$: polynomial coefficient and $\omega$

A&S Ref:

9.2.8

Referenced by:

§10.17(i), §10.17(iii), §10.17(iv), §10.41(v), §10.49(i), §10.61(v)

Permalink:

http://dlmf.nist.gov/10.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

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10.17.9 $$J_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{2}{\pi z}\right)}^{\frac{1}{2}}\left(\sin \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k}(\nu )}{z^{2k}}+\cos \omega \sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_{2k+1}(\nu )}{z^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{ph}$: phase, $\sin z$: sine function, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant, $\omega$ and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.2.11

Permalink:

http://dlmf.nist.gov/10.17.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

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17: 6.1 Special Notation

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

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18: 10.40 Asymptotic Expansions for Large Argument

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10.40.1 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: 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 and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.1

Referenced by:

§10.30(ii), §10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.45, §10.67(i), §11.6(i)

Permalink:

http://dlmf.nist.gov/10.40.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: 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 and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.3 $$I_{\nu }^{\prime }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: 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 and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.3

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.4 $$K_{\nu }^{\prime }\left(z\right)\sim -{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{b_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: 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 and $b_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.4

Referenced by:

§10.40(i), §10.40(i), §10.67(i)

Permalink:

http://dlmf.nist.gov/10.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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10.40.5 $$I_{\nu }\left(z\right)\sim \frac{{\mathrm{e}}^z}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{a_k(\nu )}{z^k}\pm \mathrm{i}{\mathrm{e}}^{\pm \nu \pi \mathrm{i}}\frac{{\mathrm{e}}^{-z}}{{(2\pi z)}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

Referenced by:

§10.40(i)

Permalink:

http://dlmf.nist.gov/10.40.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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19: 18.28 Askey–Wilson Class

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18.28.20 $$\sum _{y=0}^N{R}_n(q^{-y}+\gamma \delta q^{y+1})R_m(q^{-y}+\gamma \delta q^{y+1}){\omega }_y=h_n{\delta }_{n,m},$$ $n,m=0,1,\mathrm{\dots },N$,

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/18.28.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.21 $${\omega }_y=\frac{{(\alpha q,\beta \delta q,\gamma q,\gamma \delta q;q)}_y}{{(q,\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_y}\frac{1-\gamma \delta q^{2y+1}}{{\left(\alpha \beta q\right)}^y},$$

ⓘ

Symbols:

${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $y$: real variable, $\delta$: arbitrary small positive constant and $q$: real variable

Referenced by:

§18.28(viii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

►

18.28.22 $$h_n=\frac{{\left(\alpha \beta \right)}^{n+1}{q}^{{\left(n+1\right)}^2}}{\alpha \beta q^{2n+1}-1}\frac{{(q;q)}_n}{{(\alpha q,\beta \delta q,\gamma q;q)}_n}\frac{{(\frac{\gamma }{\alpha \beta }{q}^{-n},\frac{\delta }{\alpha }{q}^{-n},\frac{1}{\beta }{q}^{-n},\gamma \delta q;q)}_{\mathrm{\infty }}}{{(\frac{1}{\alpha \beta }{q}^{-n},\frac{\gamma \delta }{\alpha }q,\frac{\gamma }{\beta }q,\delta q;q)}_{\mathrm{\infty }}}.$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\delta$: arbitrary small positive constant, $q$: real variable, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.28.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28 and Ch.18

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18.28.23 $$R_n(q^{-y}+\gamma \delta q^{y+1};\alpha ,\beta ,\gamma ,\delta |q)=R_y(q^{-n}+\alpha \beta q^{n+1};\gamma ,\delta ,\alpha ,\beta |q),$$ $\alpha q$, $\beta \delta q$, or $\gamma q=q^{-N}$; $n,y=0,1,\mathrm{\dots },N$.

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial, $y$: real variable, $N$: positive integer, $\delta$: arbitrary small positive constant, $q$: real variable and $n$: nonnegative integer

Referenced by:

§18.28(viii), §18.28(viii), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(viii), §18.28(viii), §18.28 and Ch.18

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18.28.34 $$\underset{q\to 1}{lim}{R}_n(q^{-y}+q^{y+\gamma +\delta +1};q^{\alpha },q^{\beta },q^{\gamma },q^{\delta }|q)=R_n(y(y+\gamma +\delta +1);\alpha ,\beta ,\gamma ,\delta ).$$

ⓘ

Symbols:

$R_n(x;\alpha ,\beta ,\gamma ,\delta )$: Racah polynomial, $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$: $q$-Racah polynomial, $y$: real variable, $\delta$: arbitrary small positive constant, $q$: real variable and $n$: nonnegative integer

Source:

Koekoek et al. (2010, (14.2.24))

Referenced by:

§18.28(x), Erratum (V1.2.0) Section 18.28

Permalink:

http://dlmf.nist.gov/18.28.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §18.28(x), §18.28(x), §18.28 and Ch.18

…

20: 10.30 Limiting Forms

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10.30.4 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^z/\sqrt{2\pi z},$$ $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$,

ⓘ

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, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $\delta$: small positive constant

Permalink:

http://dlmf.nist.gov/10.30.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

►

10.30.5 $$I_{\nu }\left(z\right)\sim {\mathrm{e}}^{\pm (\nu +\frac{1}{2})\pi \mathrm{i}}{\mathrm{e}}^{-z}/\sqrt{2\pi z},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/10.30.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.30(ii), §10.30 and Ch.10

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