Debye expansions

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21: 7.17 Inverse Error Functions

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§7.17(ii) Power Series

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

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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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7.17.5 $$u=-2/\ln \left(\pi x^2\ln \left(1/x\right)\right),$$

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Defines:

$u$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E5

Encodings:

pMML, png, TeX

See also:

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

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7.17.6 $$v=\ln \left(\ln \left(1/x\right)\right)-2+\ln \pi .$$

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Defines:

$v$: expansion variable (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function and $x$: real variable

Permalink:

http://dlmf.nist.gov/7.17.E6

Encodings:

pMML, png, TeX

See also:

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

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22: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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23: 33.20 Expansions for Small $|ϵ|$

§33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(iii) Asymptotic Expansion for the Irregular Solution

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§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

24: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

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§10.41(ii) Uniform Expansions for Real Variable

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§10.41(iii) Uniform Expansions for Complex Variable

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25: 6.13 Zeros

… ►Values of $c_1$ and $c_2$ to 30D are given by MacLeod (1996b). … ►

6.13.2 $$c_k,s_k\sim \alpha +\frac{1}{\alpha }-\frac{16}{3}\frac{1}{{\alpha }^3}+\frac{1673}{15}\frac{1}{{\alpha }^5}-\frac{\mathrm{5\hspace{0.33em}07746}}{105}\frac{1}{{\alpha }^7}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $c_k$: zeros of $\mathrm{Ci}\left(x\right)$ and $s_k$: zeros of $\mathrm{Si}\left(x\right)$

Referenced by:

§6.13, §6.18(iii)

Permalink:

http://dlmf.nist.gov/6.13.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.13 and Ch.6

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26: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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

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§8.20(i) Large $z$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). ►For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii). ►

§8.20(ii) Large $p$

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27: 2.1 Definitions and Elementary Properties

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§2.1(iii) Asymptotic Expansions

… ►Symbolically, … ►For an example see (2.8.15). … ►

§2.1(iv) Uniform Asymptotic Expansions

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§2.1(v) Generalized Asymptotic Expansions

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28: 10.40 Asymptotic Expansions for Large Argument

§10.40 Asymptotic Expansions for Large Argument

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§10.40(i) Hankel’s Expansions

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Products

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§10.40(iv) Exponentially-Improved Expansions

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29: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(iii) Exponentially-Improved Expansions

… ►In this way we arrive at hyperasymptotic expansions. … ► …

30: 2.2 Transcendental Equations

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2.2.6 $$t=y^{\frac{1}{2}}\left(1+\frac{1}{4}{y}^{-1}\ln y+o\left(y^{-1}\right)\right),$$ $y\to \mathrm{\infty }$.

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Symbols:

$o\left(x\right)$: order less than, $\ln z$: principal branch of logarithm function and $y$: root

Permalink:

http://dlmf.nist.gov/2.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

►An important case is the reversion of asymptotic expansions for zeros of special functions. … ►

2.2.7 $$f(x)\sim x+f_0+f_1{x}^{-1}+f_2{x}^{-2}+\mathrm{\cdots },$$ $x\to \mathrm{\infty }$.

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Symbols:

$\sim$: Poincaré asymptotic expansion, $f(x)$: function and $f_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

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2.2.8 $$x\sim y-F_0-F_1{y}^{-1}-F_2{y}^{-2}-\mathrm{\cdots },$$ $y\to \mathrm{\infty }$,

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Symbols:

$\sim$: Poincaré asymptotic expansion, $y$: root and $F_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §2.2, §2.2 and Ch.2

►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). …