Laplace method

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1: 2.4 Contour Integrals

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§2.4(iii) Laplace’s Method

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2: 2.3 Integrals of a Real Variable

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§2.3(iii) Laplace’s Method

… ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). These references and Wong (1989, Chapter 2) also contain examples. … ►When $x\to +\mathrm{\infty }$ Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at $\alpha =0$. … ►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …

3: Bibliography N

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(11.11.16), (11.11.17), §11.11(iii).

Encodings:

BibTeX

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4: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.10 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim -\frac{1}{\pi }\sum _{k=0}^{\mathrm{\infty }}\frac{(2k)!a_k(-\lambda )}{{\nu }^{2k+1}},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

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Symbols:

${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $a_k(\lambda )$: expansion function

Sources:

Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)

Proof sketch:

Apply Laplace’s method to the integral (11.10.4).

Referenced by:

(11.11.18), (11.11.19), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E10

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \pi -\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.11 $${𝐀}_{-\nu }\left(\lambda \nu \right)\sim {\left(\frac{2}{\pi \nu }\right)}^{1/2}{\mathrm{e}}^{-\nu \mu }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}\right)}_k{b}_k(\lambda )}{{\nu }^k},$$ $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber 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, $\nu$: real or complex order, $k$: nonnegative integer, $\delta$: arbitrary small positive constant, $\lambda$: parameter and $b_k(\lambda )$: expansion function

Sources:

Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)

Proof sketch:

Derivable by applying Laplace’s method to the integral (11.10.4).

Permalink:

http://dlmf.nist.gov/11.11.E11

Encodings:

pMML, png, TeX

Clarification (effective with 1.1.2):

The constraint which was originally $\nu \to +\mathrm{\infty }$, has been extended to be $\nu \to \mathrm{\infty }$, $|\mathrm{ph}\nu |\le \frac{\pi }{2}-\delta$.

Suggested 2021-04-05 by Gergő Nemes

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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5: 11.6 Asymptotic Expansions

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6: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …

7: 2.10 Sums and Sequences

… ►By application of Laplace’s method (§2.3(iii)) and use again of (5.11.7), we obtain …

8: 2.11 Remainder Terms; Stokes Phenomenon

… ►Then by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have …

9: 15.4 Special Cases

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15.4.34 $$F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})=\sqrt{\pi }{\mathrm{e}}^{\mathrm{i}\pi a/2}\frac{2^{2a}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{3^{(3a+1)/2}}\left(\frac{1}{\mathrm{\Gamma }\left(\frac{1}{3}+a\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}+a\right)\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $a$: real or complex parameter

Proof sketch:

From §15.5(ii) one obtains that $w(a)={\mathrm{e}}^{-\mathrm{i}\pi a/2}{2}^{-2a}{3}^{(3a+1)/2}F(3a,a;2a;{\mathrm{e}}^{\mathrm{i}\pi /3})\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{1}{2}+a\right)$ is a solution of the difference equation $(3a+5)w(a+2)-6(a+1)w(a+1)+(3a+1)w(a)=0$. Two linearly independent solutions of this difference equation are $w_1(a)=1$ and $w_2(a)=\mathrm{\Gamma }\left(\frac{1}{3}+a\right)/\mathrm{\Gamma }\left(\frac{2}{3}+a\right)$. Combining (15.6.1) with Laplace’s method (see §2.4(iii)) we obtain that $w(a)\sim \sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{a^{-1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$, as $a\to \mathrm{\infty }$. Hence, $w(a)=\sqrt{\pi }\left(\frac{1}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}+\frac{w_2(a)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right)$.

Notes:

See §15.2(ii) for treatment of the case when the third parameter is a nonpositive integer.

Referenced by:

§15.4(iii), §15.4(iii), Erratum (V1.1.2) for Additions

Permalink:

http://dlmf.nist.gov/15.4.E34

Encodings:

pMML, png, TeX

Addition (effective with 1.1.2):

This equation was added.

See also:

Annotations for §15.4(iii), §15.4 and Ch.15

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10: Bibliography K

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

BibTeX

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