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51: 8.18 Asymptotic Expansions of $I_x(a,b)$

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§8.18(i) Large Parameters, Fixed $x$

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§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

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Large $a$, Fixed $b$

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Symmetric Case

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General Case

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52: 13.20 Uniform Asymptotic Approximations for Large $\mu$

§13.20 Uniform Asymptotic Approximations for Large $\mu$

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§13.20(i) Large $\mu$, Fixed $\kappa$

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§13.20(v) Large $\mu$, Other Expansions

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53: 28.4 Fourier Series

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28.4.21 $$A_{2s}^0(q)=\left(\frac{{(-1)}^{s}2}{{(s!)}^2}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+2}\right)\right)A_0^0(q),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $q=h^2$: parameter and $A_m(q)$: Fourier coefficient

A&S Ref:

20.2.29 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.4.E21

Encodings:

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See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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28.4.22 $$\begin{array}{c}{A}_{m+2s}^m(q)\\ B_{m+2s}^m(q)\end{array}\}=\left(\frac{{(-1)}^{s}m!}{s!(m+s)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q),\end{array}$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E22

Encodings:

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See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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28.4.23 $$\begin{array}{c}{A}_{m-2s}^m(q)\\ B_{m-2s}^m(q)\end{array}\}=\left(\frac{(m-s-1)!}{s!(m-1)!}{\left(\frac{q}{4}\right)}^s+O\left(q^{s+1}\right)\right)\{\begin{array}{c}{A}_m^m(q),\\ B_m^m(q).\end{array}$$

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Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $A_m(q)$: Fourier coefficient and $B_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E23

Encodings:

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See also:

Annotations for §28.4(vi), §28.4 and Ch.28

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§28.4(vii) Asymptotic Forms for Large $m$

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28.4.24 $$\frac{A_{2m}^{2n}(q)}{A_0^{2n}(q)}=\frac{{(-1)}^m}{{(m!)}^2}{\left(\frac{q}{4}\right)}^m\frac{\pi \left(1+O\left(m^{-1}\right)\right)}{w_{\text{II}}(\frac{1}{2}\pi ;a_{2n}\left(q\right),q)},$$

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Symbols:

$O\left(x\right)$: order not exceeding, $a_n\left(q\right)$: eigenvalues of Mathieu equation, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $m$: integer, $q=h^2$: parameter, $n$: integer, $w_{\text{II}}(z;a,q)$: fundamental solution and $A_m(q)$: Fourier coefficient

Permalink:

http://dlmf.nist.gov/28.4.E24

Encodings:

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See also:

Annotations for §28.4(vii), §28.4 and Ch.28

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54: 11.9 Lommel Functions

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11.9.1 $$\frac{d^{2}w}{{dz}^2}+\frac{1}{z}\frac{dw}{dz}+\left(1-\frac{{\nu }^2}{z^2}\right)w=z^{\mu -1}$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex order

Referenced by:

§11.9(i), §11.9(i), §11.9(iii)

Permalink:

http://dlmf.nist.gov/11.9.E1

Encodings:

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See also:

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

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11.9.2 $$w=s_{\mu ,\nu }\left(z\right)+A{J}_{\nu }\left(z\right)+B{Y}_{\nu }\left(z\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $s_{\mu ,\nu }\left(z\right)$: Lommel function, $z$: complex variable, $\nu$: real or complex order, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/11.9.E2

Encodings:

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See also:

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

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11.9.3 $$s_{\mu ,\nu }\left(z\right)=z^{\mu +1}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{z^{2k}}{a_{k+1}(\mu ,\nu )},$$

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Defines:

$s_{\mu ,\nu }\left(z\right)$: Lommel function

Symbols:

$z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer and $a_k(\mu ,\nu )$: expansion function

Permalink:

http://dlmf.nist.gov/11.9.E3

Encodings:

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See also:

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

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§11.9(iii) Asymptotic Expansion

… ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). …

55: Bibliography T

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Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

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External Links:

FullText

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

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External Links:

FullText, MathReview (G. Meinardus), ZentralBlatt

Referenced by:

§8.25(iii).

Encodings:

BibTeX

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Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

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External Links:

ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

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External Links:

ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§13.8(iv).

Encodings:

BibTeX

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Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

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External Links:

MathReview (H. Bückner), ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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56: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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§8.11(i) Large $z$, Fixed $a$

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§8.11(ii) Large $a$, Fixed $z$

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§8.11(iii) Large $a$, Fixed $z/a$

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57: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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§13.21(i) Large $\kappa$, Fixed $\mu$

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§13.21(ii) Large $\kappa$, $0\le \mu \le (1-\delta )\kappa$

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§13.21(iv) Large $\kappa$, Other Expansions

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58: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

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The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

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External Links:

ISSN 0021-9045, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.

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External Links:

ISSN 0196-8858, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.

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External Links:

ISSN 1660-5446, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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59: 2.6 Distributional Methods

… ►To derive an asymptotic expansion of $𝒮f\left(z\right)$ for large values of $|z|$, with , we assume that $f(t)$ possesses an asymptotic expansion of the form … ►On substituting (2.6.15) into (2.6.26) and interchanging the order of integration, the right-hand side of (2.6.26) becomes … ►The Riemann–Liouville fractional integral of order $\mu$ is defined by …We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

60: 36.12 Uniform Approximation of Integrals

… ►where $k$ is a large real parameter and $𝐲=\{y_1,y_2,\mathrm{\dots }\}$ is a set of additional (nonasymptotic) parameters. … ►The leading-order uniform asymptotic approximation is given by ►

36.12.3 $$I(𝐲,k)=\frac{\exp \left(\mathrm{i}kA(𝐲)\right)}{k^{1/(K+2)}}\sum _{m=0}^K\frac{a_m(𝐲)}{k^{m/(K+2)}}\left({\delta }_{m,0}-\left(1-{\delta }_{m,0}\right)\mathrm{i}\frac{\partial }{\partial z_m}\right){\mathrm{\Psi }}_K\left(𝐳(𝐲;k)\right)\left(1+O\left(\frac{1}{k}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${\delta }_{j,k}$: Kronecker delta, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: integer, $k$: variable, $K$: codimension, $I(𝐲,k)$: oscillatory integral, $a_m(𝐲)$: function, $A(𝐲)$: function, $𝐳(𝐲,k)$: function and $z_m(𝐲,k)$: function

Permalink:

http://dlmf.nist.gov/36.12.E3

Encodings:

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See also:

Annotations for §36.12(i), §36.12 and Ch.36

… ►Correspondence between the $u_j(𝐲)$ and the $t_j(𝐱)$ is established by the order of critical points along the real axis when $𝐲$ and $𝐱$ are such that these critical points are all real, and by continuation when some or all of the critical points are complex. … ►For example, the diffraction catastrophe ${\mathrm{\Psi }}_2(x,y;k)$ defined by (36.2.10), and corresponding to the Pearcey integral (36.2.14), can be approximated by the Airy function ${\mathrm{\Psi }}_1\left(\xi (x,y;k)\right)$ when $k$ is large, provided that $x$ and $y$ are not small. …