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1: 16.5 Integral Representations and Integrals

… ►In this event, the formal power-series expansion of the left-hand side (obtained from (16.2.1)) is the asymptotic expansion of the right-hand side as $z\to 0$ in the sector $|\mathrm{ph}\left(-z\right)|\le (p+1-q-\delta )\pi /2$, where $\delta$ is an arbitrary small positive constant. …

2: 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

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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

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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

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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

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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

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See also:

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

…

3: 9.1 Special Notation

… ► ►►►►►

$k$	nonnegative integer, except in §9.9(iii).
$x$	real variable.
$z(=x+\mathrm{i}y)$	complex variable.
$\delta$	arbitrary small positive constant.
…

…

4: 5.1 Special Notation

… ► ►►►►►►►

$j,m,n$	nonnegative integers.
…
$x,y$	real variables.
$z=x+\mathrm{i}y$	complex variable.
$a,b,q,s,w$	real or complex variables with .
$\delta$	arbitrary small positive constant.
…
primes	derivatives with respect to the variable.

…

5: 13.1 Special Notation

… ► ►►►►►

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

…

6: 13.19 Asymptotic Expansions for Large Argument

… ►

13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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 $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

… ►

13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric 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, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

…

7: 13.7 Asymptotic Expansions for Large Argument

… ►

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:

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See also:

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

… ►

13.7.3 $$U(a,b,z)\sim z^{-a}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_s{\left(a-b+1\right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

ⓘ

Symbols:

$U(a,b,z)$: Kummer 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, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

A&S Ref:

13.5.2

Referenced by:

§12.9(i), §13.12, §13.2(i)

Permalink:

http://dlmf.nist.gov/13.7.E3

Encodings:

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See also:

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

… ►

13.7.13

ⓘ

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, $\mathrm{ph}$: phase, $m$: integer, $n$: nonnegative integer, $z$: complex variable, $\delta$: small positive constant and $R_n(a,b,z)$: remainder

Permalink:

http://dlmf.nist.gov/13.7.E13

Encodings:

pMML, png, TeX

See also:

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

…

8: 9.13 Generalized Airy Functions

… ►

9.13.9 $$A_n\left(z\right)=\sqrt{p/\pi }\sin \left(p\pi \right)z^{-n/4}{\mathrm{e}}^{-\zeta }\left(1+O\left({\zeta }^{-1}\right)\right),$$ $|\mathrm{ph}z|\le 3p\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (13))

Permalink:

http://dlmf.nist.gov/9.13.E9

Encodings:

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See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.10

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $A_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (14)–(15))

Permalink:

http://dlmf.nist.gov/9.13.E10

Encodings:

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See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.11 $$B_n\left(z\right)={\pi }^{-1/2}{z}^{-n/4}{\mathrm{e}}^{\zeta }\left(1+O\left({\zeta }^{-1}\right)\right),$$ $|\mathrm{ph}z|\le p\pi -\delta$,

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{ph}$: phase, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (16))

Permalink:

http://dlmf.nist.gov/9.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.13(i), §9.13 and Ch.9

►

9.13.12

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $B_n\left(z\right)$: generalized Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $\delta$: small positive constant, $n$: parameter, $p$: variable and $\zeta$: variable

Source:

Swanson and Headley (1967, (17)–(18))

Permalink:

http://dlmf.nist.gov/9.13.E12

Encodings:

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See also:

Annotations for §9.13(i), §9.13 and Ch.9

…

9: 11.13 Methods of Computation

… ►For numerical purposes the most convenient of the representations given in §11.5, at least for real variables, include the integrals (11.5.2)–(11.5.5) for ${𝐊}_{\nu }\left(z\right)$ and ${𝐌}_{\nu }\left(z\right)$. …Other integrals that appear in §11.5(i) have highly oscillatory integrands unless $z$ is small. ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. The solution ${𝐊}_{\nu }\left(x\right)$ needs to be integrated backwards for small $x$, and either forwards or backwards for large $x$ depending whether or not $\nu$ exceeds $\frac{1}{2}$. …

10: 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

…