asymptotic expansions

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31: 33.12 Asymptotic Expansions for Large $\eta$

§33.12 Asymptotic Expansions for Large $\eta$

… ►For asymptotic expansions of $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \pm \mathrm{\infty }$ see Temme (2015, Chapter 31). ►

§33.12(ii) Uniform Expansions

… ►Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \mathrm{\infty }$. … ►

32: 7.12 Asymptotic Expansions

§7.12 Asymptotic Expansions

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§7.12(i) Complementary Error Function

… ►

§7.12(ii) Fresnel Integrals

►The asymptotic expansions of $C\left(z\right)$ and $S\left(z\right)$ are given by (7.5.3), (7.5.4), and … ►

§7.12(iii) Goodwin–Staton Integral

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33: 28.26 Asymptotic Approximations for Large $q$

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►

28.26.5 $${\mathrm{Gc}}_m(z,h)\sim \frac{\sinh z}{{\cosh }^{2}z}\left(\frac{s^2+3}{2^{5}h}+\frac{1}{2^9{h}^2}\left(s^3+3s+\frac{4s^3+44s}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(5s^4+34s^2+9-\frac{s^6-47s^4+667s^2+2835}{12{\cosh }^{2}z}+\frac{s^6+505s^4+12139s^2+10395}{12{\cosh }^{4}z}\right)\right)+\mathrm{\cdots }.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.10 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. …

34: 5.15 Polygamma Functions

… ►This includes asymptotic expansions: compare §§2.1(ii)–2.1(iii). … ►

5.15.8 $${\psi }^{\prime }\left(z\right)\sim \frac{1}{z}+\frac{1}{2z^2}+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{z^{2k+1}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.12

Referenced by:

§5.15

Permalink:

http://dlmf.nist.gov/5.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

►

5.15.9 $${\psi }^{(n)}\left(z\right)\sim {(-1)}^{n-1}\left(\frac{(n-1)!}{z^n}+\frac{n!}{2z^{n+1}}+\sum _{k=1}^{\mathrm{\infty }}\frac{(2k+n-1)!}{(2k)!}\frac{B_{2k}}{z^{2k+n}}\right).$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.11

Referenced by:

§5.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/5.15.E9

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

Suggested 2021-09-09 by Calvin Khor

See also:

Annotations for §5.15 and Ch.5

…

35: 10.20 Uniform Asymptotic Expansions for Large Order

§10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

… ► ►

§10.20(iii) Double Asymptotic Properties

►For asymptotic properties of the expansions (10.20.4)–(10.20.6) with respect to large values of $z$ see §10.41(v).

36: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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§9.7(v) Exponentially-Improved Expansions

… ►

37: 9.15 Mathematical Applications

… ►Airy functions play an indispensable role in the construction of uniform asymptotic expansions for contour integrals with coalescing saddle points, and for solutions of linear second-order ordinary differential equations with a simple turning point. …

38: 7.17 Inverse Error Functions

… ►

§7.17(iii) Asymptotic Expansion of $\mathrm{inverfc}x$ for Small $x$

… ►

7.17.3 $$\mathrm{inverfc}x\sim u^{-1/2}+a_2{u}^{3/2}+a_3{u}^{5/2}+a_4{u}^{7/2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\mathrm{inverfc}x$: inverse complementary error function, $x$: real variable, $a_{\mathrm{i}}$: coefficients and $u$: expansion variable

Referenced by:

§7.17(iii), §7.17(iii)

Permalink:

http://dlmf.nist.gov/7.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.17(iii), §7.17 and Ch.7

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39: 13.7 Asymptotic Expansions for Large Argument

§13.7 Asymptotic Expansions for Large Argument

… ►

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

… ►

§13.7(ii) Error Bounds

… ►

§13.7(iii) Exponentially-Improved Expansion

… ►For extensions to hyperasymptotic expansions see Olde Daalhuis and Olver (1995a).

40: 2.11 Remainder Terms; Stokes Phenomenon

… ►

§2.11(i) Numerical Use of Asymptotic Expansions

… ►

2.11.4 $$I(10)\approx -\mathrm{0.00053\hspace{0.33em}18}+\mathrm{0.00000\hspace{0.33em}48}-\mathrm{0.00000\hspace{0.33em}01}=-\mathrm{0.00052\hspace{0.33em}71}.$$

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Symbols:

$\sim$: Poincaré asymptotic expansion and $I(m)$: integral

Referenced by:

§2.11(i), Erratum (V1.0.12) for Equation (2.11.4), Erratum (V1.0.12) for Equation (2.11.4)

Permalink:

http://dlmf.nist.gov/2.11.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Because this is not an asymptoticexpansion, the symbol $\sim$ that was used originally is incorrect and has been replaced with $\approx$, together with a slight change of wording.

See also:

Annotations for §2.11(i), §2.11 and Ch.2

… ►

§2.11(iii) Exponentially-Improved Expansions

… ► … ►

2.11.29 $$W_{\kappa ,\mu }\left(z\right)\sim \sum _{n=0}^{\mathrm{\infty }}{a}_n,$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion and $a_n$: coefficients

Permalink:

http://dlmf.nist.gov/2.11.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(vi), §2.11 and Ch.2

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