asymptotic expansions of integrals

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1: 6.12 Asymptotic Expansions

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§6.12(i) Exponential and Logarithmic Integrals

… ►For the function $\chi$ see §9.7(i). … ►

§6.12(ii) Sine and Cosine Integrals

… ► … ►

2: 16.25 Methods of Computation

… ►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …

3: 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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4: 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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5: 7.12 Asymptotic Expansions

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§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 ►

7.12.2 $$\mathrm{f}\left(z\right)\sim \frac{1}{\pi z}\sum _{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{{\left(\frac{1}{2}\right)}_{2m}}{{(\pi z^2/2)}^{2m}},$$

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Symbols:

$\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

A&S Ref:

7.3.27

Referenced by:

§7.12(ii), §7.12(ii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/7.12.E2

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

Previously the RHS of this equation was written as $\frac{1}{\pi z}{\sum }_{m=0}^{\mathrm{\infty }}{(-1)}^m\frac{1\cdot 3\cdot 5\mathrm{\cdots }(4m-1)}{{(\pi z^2)}^{2m}}$. We have rewritten the sum more concisely using Pochhammer’s symbol.

Suggested 2014-03-13 by Giorgos Karagounis

See also:

Annotations for §7.12(ii), §7.12 and Ch.7

… ►They are bounded by $|\csc \left(4\mathrm{ph}z\right)|$ times the first neglected terms when . … ►

§7.12(iii) Goodwin–Staton Integral

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6: 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. …

7: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

8: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►

§2.3(ii) Watson’s Lemma

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2.3.12 $${\int }_0^{\mathrm{\infty }}f(xt)q(t)dt\sim \sum _{s=0}^{\mathrm{\infty }}ℳf\left(\frac{s+\lambda }{\mu }\right)\frac{a_s}{x^{(s+\lambda )/\mu }},$$ $x\to +\mathrm{\infty }$,

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $\sim$: Poincaré asymptotic expansion, $dx$: differential of $x$ , $\int$: integral, $f(x)$: function, $a$: positive constant, $q(t)$: function, $\lambda$: positive constant and $\mu$: positive constant

Referenced by:

§2.3(ii), §2.3(ii)

Permalink:

http://dlmf.nist.gov/2.3.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §2.3(ii), §2.3 and Ch.2

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§2.3(iii) Laplace’s Method

… ►Then …

9: 2.6 Distributional Methods

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2.6.3 $${\int }_0^{\mathrm{\infty }}\frac{t^{-s-(1/3)}}{x+t}dt,$$ $s=1,2,3,\mathrm{\dots }$.

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Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/2.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(i), §2.6 and Ch.2

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§2.6(ii) Stieltjes Transform

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§2.6(iii) Fractional Integrals

… ►If both $f$ and $g$ in (2.6.34) have asymptotic expansions of the form (2.6.9), then the distribution method can also be used to derive an asymptotic expansion of the convolution $f\ast g$; see Li and Wong (1994). … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: …

10: 7.13 Zeros

… ►As $n\to \mathrm{\infty }$ the $x_n$ and $y_n$ corresponding to the zeros of $C\left(z\right)$ satisfy … ►For an asymptotic expansion of the zeros of ${\int }_0^z\exp \left(\frac{1}{2}\pi \mathrm{i}{t}^2\right)dt$ ($=ℱ\left(0\right)-ℱ\left(z\right)$ $=C\left(z\right)+\mathrm{i}S\left(z\right)$) see Tuẑilin (1971).