small k,k′

SearchAdvancedHelp

(0.009 seconds)

11—20 of 70 matching pages

11: 10.40 Asymptotic Expansions for Large Argument

… ►

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

►

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

►

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

►

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

… ►

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

…

12: 11.11 Asymptotic Expansions of Anger–Weber Functions

… ►

11.11.8 $${𝐀}_{\nu }\left(\lambda \nu \right)\sim \frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$,

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Referenced by:

(11.11.18), (11.11.19), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which originally was $\nu \to \mathrm{\infty }$, , has been replaced to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

ⓘ

Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

… ►

11.11.11 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_k{b}_k(\lambda )}{{\nu }^k},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Sources:

Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)

Proof sketch:

Derivable by applying Laplace’s method to the integral (11.10.4).

Permalink:

http://dlmf.nist.gov/11.11.E11

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

…

13: 26.6 Other Lattice Path Numbers

… ►

26.6.8 $$\sum _{n,k=1}^{\mathrm{\infty }}N(n,k)x^n{y}^k=\frac{1-x-xy-\sqrt{{(1-x-xy)}^2-4x^{2}y}}{2x},$$

ⓘ

Symbols:

$k$: nonnegative integer, $n$: nonnegative integer, $N(n,k)$: Narayana number, $x$: small and $y$: small

Permalink:

http://dlmf.nist.gov/26.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §26.6(ii), §26.6 and Ch.26

…

14: 11.9 Lommel Functions

… ►

11.9.9 $$S_{\mu ,\nu }\left(z\right)\sim z^{\mu -1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k{a}_k(-\mu ,\nu )z^{-2k},$$ $z\to \mathrm{\infty }$, .

ⓘ

Symbols:

$S_{\mu ,\nu }\left(z\right)$: Lommel 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, $\delta$: arbitrary small positive constant and $a_k(\mu ,\nu )$: expansion function

Referenced by:

§11.9(iii), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.9(iii), §11.9 and Ch.11

…

15: 10.19 Asymptotic Expansions for Large Order

… ►

$$Y_{\nu }\left(\nu \mathrm{sech}\alpha \right)\sim -\frac{{\mathrm{e}}^{\nu (\alpha -\tanh \alpha )}}{{(\frac{1}{2}\pi \nu \tanh \alpha )}^{\frac{1}{2}}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{U_k(\coth \alpha )}{{\nu }^k},$$

… ►

$$Y_{\nu }^{\prime }\left(\nu \mathrm{sech}\alpha \right)\sim {\left(\frac{\sinh \left(2\alpha \right)}{\pi \nu }\right)}^{\frac{1}{2}}{\mathrm{e}}^{\nu (\alpha -\tanh \alpha )}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{V_k(\coth \alpha )}{{\nu }^k}.$$

… ►

$$J_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}-\mathrm{i}\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

►

$$Y_{\nu }\left(\nu \sec \beta \right)\sim {\left(\frac{2}{\pi \nu \tan \beta }\right)}^{\frac{1}{2}}\left(\sin \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k}(\mathrm{i}\cot \beta )}{{\nu }^{2k}}+\mathrm{i}\cos \xi \sum _{k=0}^{\mathrm{\infty }}\frac{U_{2k+1}(\mathrm{i}\cot \beta )}{{\nu }^{2k+1}}\right),$$

… ►In these expansions $U_k(p)$ and $V_k(p)$ are the polynomials in $p$ of degree $3k$ defined in §10.41(ii). …

16: 9.7 Asymptotic Expansions

… ►

9.7.5 $$\mathrm{Ai}\left(z\right)\sim \frac{{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

(9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.6 $${\mathrm{Ai}}^{\prime }\left(z\right)\sim -\frac{z^{1/4}{\mathrm{e}}^{-\zeta }}{2\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.07), pp. 392 and 413)

Referenced by:

§9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)

Permalink:

http://dlmf.nist.gov/9.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.7 $$\mathrm{Bi}\left(z\right)\sim \frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

(9.10.5), (9.10.7), §9.7(iii), §9.7(iii), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.8 $${\mathrm{Bi}}^{\prime }\left(z\right)\sim \frac{z^{1/4}{\mathrm{e}}^{\zeta }}{\sqrt{\pi }}\sum _{k=0}^{\mathrm{\infty }}\frac{v_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $v_s$: expansion coefficient

Source:

Olver (1997b, (1.16), pp. 393 and 414)

Referenced by:

§9.7(iii), §9.7(iii), Erratum (V1.0.17) for Subsection 9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

►

9.7.9 $$\mathrm{Ai}\left(-z\right)\sim \frac{1}{\sqrt{\pi }{z}^{1/4}}\left(\cos \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k}}{{\zeta }^{2k}}+\sin \left(\zeta -\frac{1}{4}\pi \right)\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{u_{2k+1}}{{\zeta }^{2k+1}}\right),$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: 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, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Source:

Olver (1997b, (1.08), pp. 392 and 413)

Referenced by:

§9.7(iii)

Permalink:

http://dlmf.nist.gov/9.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §9.7(ii), §9.7 and Ch.9

…

17: 9.12 Scorer Functions

… ►

9.12.25 $$\mathrm{Gi}\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

►

9.12.26 $${\mathrm{Gi}}^{\prime }\left(z\right)\sim -\frac{1}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k+1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Refer to §2.1(ii).

Permalink:

http://dlmf.nist.gov/9.12.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

►

9.12.27 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.31)

Permalink:

http://dlmf.nist.gov/9.12.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

►

9.12.28 $${\mathrm{Hi}}^{\prime }\left(z\right)\sim \frac{1}{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k+1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$.

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

Refer to §2.1(ii).

Permalink:

http://dlmf.nist.gov/9.12.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

… ►

9.12.29 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k}+\frac{{\mathrm{e}}^{\zeta }}{\sqrt{\pi }{z}^{1/4}}\sum _{k=0}^{\mathrm{\infty }}\frac{u_k}{{\zeta }^k},$$ $|\mathrm{ph}z|\le \pi -\delta$.

ⓘ

Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant, $\zeta$: change of variable and $u_s$: expansion coefficient

Permalink:

http://dlmf.nist.gov/9.12.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

…

18: 10.41 Asymptotic Expansions for Large Order

… ►

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

►

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

…

19: 15.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$k,\mathrm{\ell },m,n$	integers.
…
$\delta$	arbitrary small positive constant.
…

…

20: 19.12 Asymptotic Approximations

… ►They are useful primarily when $(1-k)/(1-\sin \varphi )$ is either small or large compared with 1. …