36 Integrals with Coalescing SaddlesProperties36.7 Zeros36.9 Integral Identities

§36.8 Convergent Series Expansions

ⓘ

Keywords:

Pearcey integral, canonical integrals, convergent series, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral

Notes:

See Connor (1973) and Connor et al. (1983). For (36.8.1), in the integral (36.2.4) retain the highest power of $t$ in (36.2.1) in the exponent, expand the rest of the exponential as a power series in $t$, and evaluate the resulting integrals in terms of gamma functions. For (36.8.3), in the integral (36.2.5) with the polynomial (36.2.3) expand the $z$-dependent part of the exponential in powers of $z$, and then repeatedly use the differential equation (9.2.1) to express higher derivatives of the Airy function in terms of $\mathrm{Ai}$ and ${\mathrm{Ai}}^{\prime }$. For (36.8.4), in the integral (36.2.5) with the polynomial (36.2.2) expand the $z$-dependent part of the exponential in powers of $z$, and then repeatedly use (9.2.1), and (36.2.20).

Referenced by:

§36.15(i), Erratum (V1.0.19) for Notation

Permalink:

http://dlmf.nist.gov/36.8

See also:

Annotations for Ch.36

36.8.1		${\mathrm{\Psi }}_K\left(𝐱\right)$	$={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}\exp \left(\mathrm{i}{\displaystyle \frac{\pi (2n+1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{2n+1}{K+2}}\right)a_{2n}(𝐱),$
	$K$ even,
		${\mathrm{\Psi }}_K\left(𝐱\right)$	$={\displaystyle \frac{2}{K+2}}{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\mathrm{i}}^n\cos \left({\displaystyle \frac{\pi (n(K+1)-1)}{2(K+2)}}\right)\mathrm{\Gamma }\left({\displaystyle \frac{n+1}{K+2}}\right)a_n(𝐱),$
	$K$ odd,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $m$: integer, $K$: codimension and $a_n(𝐱)$: coefficients Referenced by: §36.8 Permalink: http://dlmf.nist.gov/36.8.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.8 and Ch.36

where

36.8.2		$a_0(𝐱)$	$=1,$
		$a_{n+1}(𝐱)$	$={\displaystyle \frac{\mathrm{i}}{n+1}}{\displaystyle \sum _{p=0}^{\min (n,K-1)}}(p+1)x_{p+1}{a}_{n-p}(𝐱),$
	$n=0,1,2,\mathrm{\dots }$.
ⓘ Defines: $a_n(𝐱)$: coefficients (locally) Symbols: $\mathrm{i}$: imaginary unit, $m$: integer, $K$: codimension and $x_{\mathrm{i}}$: real parameter Permalink: http://dlmf.nist.gov/36.8.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §36.8 and Ch.36

For multinomial power series for ${\mathrm{\Psi }}_K\left(𝐱\right)$, see Connor and Curtis (1982).

36.8.3		$$\frac{3^{2/3}}{4{\pi }^2}{\mathrm{\Psi }}^{(\mathrm{H})}\left(3^{1/3}𝐱\right)=\begin{array}{l}\mathrm{Ai}\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=0}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)c_n(y)}{n!}\\ +\mathrm{Ai}\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{c_n(x)d_n(y)}{n!}\\ +{\mathrm{Ai}}^{\prime }\left(x\right)\mathrm{Ai}\left(y\right)\sum _{n=2}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)c_n(y)}{n!}\\ +{\mathrm{Ai}}^{\prime }\left(x\right){\mathrm{Ai}}^{\prime }\left(y\right)\sum _{n=1}^{\mathrm{\infty }}{(-3^{-1/3}\mathrm{i}z)}^n\frac{d_n(x)d_n(y)}{n!},\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter, $c_n(t)$: coefficients and $d_n(t)$: coefficients Referenced by: §36.8 Permalink: http://dlmf.nist.gov/36.8.E3 Encodings: TeX, pMML, png See also: Annotations for §36.8 and Ch.36

and

36.8.4		$${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)=2{\pi }^2{\left(\frac{2}{3}\right)}^{2/3}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(-\mathrm{i}{(2/3)}^{2/3}z\right)}^n}{n!}\mathrm{\Re }\left(f_n(\frac{x+\mathrm{i}y}{{12}^{1/3}},\frac{x-\mathrm{i}y}{{12}^{1/3}})\right),$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter, $m$: integer, $x$: real parameter and $f_n$: function Referenced by: §36.8 Permalink: http://dlmf.nist.gov/36.8.E4 Encodings: TeX, pMML, png See also: Annotations for §36.8 and Ch.36

where

36.8.5		$$f_n(\zeta ,\overline{\zeta })=\begin{array}{l}{c}_n(\zeta )c_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+c_n(\zeta )d_n(\overline{\zeta })\mathrm{Ai}\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right)\\ +d_n(\zeta )c_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right)\mathrm{Bi}\left(\overline{\zeta }\right)+d_n(\zeta )d_n(\overline{\zeta }){\mathrm{Ai}}^{\prime }\left(\zeta \right){\mathrm{Bi}}^{\prime }\left(\overline{\zeta }\right),\end{array}$$
ⓘ Defines: $f_n$: function (locally) Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\overline{z}$: complex conjugate, $m$: integer, $c_n(t)$: coefficients and $d_n(t)$: coefficients Permalink: http://dlmf.nist.gov/36.8.E5 Encodings: TeX, pMML, png See also: Annotations for §36.8 and Ch.36

and

36.8.6		$c_0(t)$	$=1,$
		$d_0(t)$	$=0,$
		$c_{n+1}(t)$	$=c_n^{\prime }(t)+t{d}_n(t),$
		$d_{n+1}(t)$	$=c_n(t)+d_n^{\prime }(t).$
ⓘ Defines: $c_n(t)$: coefficients (locally) and $d_n(t)$: coefficients (locally) Symbols: $m$: integer and $t$: variable Permalink: http://dlmf.nist.gov/36.8.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §36.8 and Ch.36

36.7 Zeros36.9 Integral Identities

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