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41: 8.18 Asymptotic Expansions of $I_x(a,b)$

§8.18 Asymptotic Expansions of $I_x(a,b)$

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§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

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Symmetric Case

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General Case

… ►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

42: 11.11 Asymptotic Expansions of Anger–Weber Functions

§11.11 Asymptotic Expansions of Anger–Weber Functions

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§11.11(i) Large $|z|$, Fixed $\nu$

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11.11.4 $${𝐀}_{\nu }\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{F_k(\nu )}{z^{2k}}-\frac{\nu }{\pi z^2}\sum _{k=0}^{\mathrm{\infty }}\frac{G_k(\nu )}{z^{2k}}.$$

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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, $z$: complex variable, $\nu$: real or complex order, $k$: nonnegative integer, $F_k(\nu )$: expansion functions and $G_k(\nu )$: expansion functions

Permalink:

http://dlmf.nist.gov/11.11.E4

Encodings:

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See also:

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

… ► … ►For an extension of (11.11.17) (and (11.11.16)) into a complete asymptotic expansion, see Nemes (2020). …

43: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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44: 9.8 Modulus and Phase

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§9.8(iv) Asymptotic Expansions

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9.8.20 $$M^2\left(x\right)\sim \frac{1}{\pi {(-x)}^{1/2}}\sum _{k=0}^{\mathrm{\infty }}\frac{1\cdot 3\cdot 5\mathrm{\cdots }(6k-1)}{k!{(96)}^k}\frac{1}{x^{3k}},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $M\left(z\right)$: Airy modulus function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii)

Referenced by:

§9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E20

Encodings:

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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9.8.22 $$\theta \left(x\right)\sim \frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1+\frac{5}{32}\frac{1}{x^3}+\frac{1105}{6144}\frac{1}{x^6}+\frac{82825}{65536}\frac{1}{x^9}+\frac{\mathrm{12820\hspace{0.33em}31525}}{\mathrm{587\hspace{0.33em}20256}}\frac{1}{x^{12}}+\mathrm{\cdots }\right),$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\theta \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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9.8.23 $$\varphi \left(x\right)\sim -\frac{\pi }{4}+\frac{2}{3}{(-x)}^{3/2}\left(1-\frac{7}{32}\frac{1}{x^3}-\frac{1463}{6144}\frac{1}{x^6}-\frac{\mathrm{4\hspace{0.33em}95271}}{\mathrm{3\hspace{0.33em}27680}}\frac{1}{x^9}-\frac{\mathrm{2065\hspace{0.33em}30429}}{\mathrm{83\hspace{0.33em}88608}}\frac{1}{x^{12}}-\mathrm{\cdots }\right).$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\varphi \left(z\right)$: Airy phase function and $x$: real variable

Proof sketch:

Combine (9.8.9)–(9.8.12) with §10.18(iii). See also Nemes (2021, Theorem 2.1 and p. 4333).

Referenced by:

§9.8(iv), §9.8(iv), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv)

Permalink:

http://dlmf.nist.gov/9.8.E23

Encodings:

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See also:

Annotations for §9.8(iv), §9.8 and Ch.9

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45: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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30.9.1 $${\lambda }_n^m\left({\gamma }^2\right)\sim -{\gamma }^2+\gamma q+{\beta }_0+{\beta }_1{\gamma }^{-1}+{\beta }_2{\gamma }^{-2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and ${\beta }_n$: coeffients

A&S Ref:

21.7.6

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §30.9(i), §30.9 and Ch.30

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

30.9.4 $${\lambda }_n^m\left({\gamma }^2\right)\sim 2q|\gamma |+c_0+c_1{|\gamma |}^{-1}+c_2{|\gamma |}^{-2}+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, ${\gamma }^2$: real parameter, $q$ and $c_n$: coeffients

A&S Ref:

21.8.2

Referenced by:

item

Permalink:

http://dlmf.nist.gov/30.9.E4

Encodings:

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See also:

Annotations for §30.9(ii), §30.9 and Ch.30

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …

46: 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. …

47: Bibliography Q

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Uniform asymptotic expansions of a double integral: Coalescence of two stationary points. Proc. Roy. Soc. London Ser. A 456, pp. 407–431.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

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Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.

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External Links:

ISSN 1617-9447, FullText, MathReview (C. M. Joshi), ZentralBlatt (N. M. Temme)

Referenced by:

§18.24.

Encodings:

BibTeX

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48: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

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49: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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10.21.21 $$j_{\nu ,m}^{\prime \prime }\sim c-\frac{\mu +7}{8c}-\frac{28{\mu }^2+424\mu +1724}{3{(8c)}^3}-\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $m$: integer, $\nu$: complex parameter and $c$: zero of cylinder function

Permalink:

http://dlmf.nist.gov/10.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vi), §10.21 and Ch.10

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§10.21(vii) Asymptotic Expansions for Large Order

… ►For derivations and further information, including extensions to uniform asymptotic expansions, see Olver (1954, 1960). …

50: 9.12 Scorer Functions

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§9.12(viii) Asymptotic Expansions

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9.12.25 $$\mathrm{Gi}\left(z\right)\sim \frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$,

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Symbols:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E25

Encodings:

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See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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9.12.27 $$\mathrm{Hi}\left(z\right)\sim -\frac{1}{\pi z}\sum _{k=0}^{\mathrm{\infty }}\frac{(3k)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}\left(-z\right)|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Proof sketch:

See Olver (1997b, pp. 431–432).

Referenced by:

(9.12.31)

Permalink:

http://dlmf.nist.gov/9.12.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

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