§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 $\operatorname{Ai}$ and $\operatorname{Ai}'$ . 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		$\displaystyle\Psi_{K}\left(\mathbf{x}\right)$	$\displaystyle=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}\exp\left(i\dfrac{\pi(2n% +1)}{2(K+2)}\right)\Gamma\left(\dfrac{2n+1}{K+2}\right)a_{2n}(\mathbf{x}),$
	$K$ even,
		$\displaystyle\Psi_{K}\left(\mathbf{x}\right)$	$\displaystyle=\dfrac{2}{K+2}\sum\limits_{n=0}^{\infty}i^{n}\cos\left(\dfrac{% \pi(n(K+1)-1)}{2(K+2)}\right)\Gamma\left(\dfrac{n+1}{K+2}\right)a_{n}(\mathbf{% x}),$
	$K$ odd,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$ : canonical integral function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $m$ : integer, $K$ : codimension and $a_{n}(\mathbf{x})$ : 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		$\displaystyle a_{0}(\mathbf{x})$	$\displaystyle=1,$
		$\displaystyle a_{n+1}(\mathbf{x})$	$\displaystyle=\dfrac{i}{n+1}\sum_{p=0}^{\min(n,K-1)}(p+1)x_{p+1}a_{n-p}(% \mathbf{x}),$
	$n=0,1,2,\dots$ .
ⓘ Defines: $a_{n}(\mathbf{x})$ : coefficients (locally) Symbols: $\mathrm{i}$ : imaginary unit, $m$ : integer, $K$ : codimension and $x_{i}$ : 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 $\Psi_{K}\left(\mathbf{x}\right)$ , see Connor and Curtis (1982).

36.8.3		$\dfrac{3^{2/3}}{4\pi^{2}}\Psi^{(\mathrm{H})}\left(3^{1/3}\mathbf{x}\right)=% \operatorname{Ai}\left(x\right)\operatorname{Ai}\left(y\right)\sum\limits_{n=0% }^{\infty}(-3^{-1/3}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}\left% (x\right)\operatorname{Ai}'\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}% iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)% \operatorname{Ai}\left(y\right)\sum\limits_{n=2}^{\infty}(-3^{-1/3}iz)^{n}% \dfrac{d_{n}(x)c_{n}(y)}{n!}+\operatorname{Ai}'\left(x\right)\operatorname{Ai}% '\left(y\right)\sum\limits_{n=1}^{\infty}(-3^{-1/3}iz)^{n}\dfrac{d_{n}(x)d_{n}% (y)}{n!},$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$ : 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		$\Psi^{(\mathrm{E})}\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{% 2/3}\sum\limits_{n=0}^{\infty}\dfrac{\left(-i(2/3)^{2/3}z\right)^{n}}{n!}\Re% \left(f_{n}\left(\dfrac{x+iy}{12^{1/3}},\dfrac{x-iy}{12^{1/3}}\right)\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$ : elliptic umbilic canonical integral function, $!$ : factorial (as in $n!$ ), $\mathrm{i}$ : imaginary unit, $\Re$ : 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		$f_{n}(\zeta,\overline{\zeta})=c_{n}(\zeta)c_{n}(\overline{\zeta})\operatorname% {Ai}\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}\right)+c_{n}(% \zeta)d_{n}(\overline{\zeta})\operatorname{Ai}\left(\zeta\right)\operatorname{% Bi}'\left(\overline{\zeta}\right)+d_{n}(\zeta)c_{n}(\overline{\zeta})% \operatorname{Ai}'\left(\zeta\right)\operatorname{Bi}\left(\overline{\zeta}% \right)+d_{n}(\zeta)d_{n}(\overline{\zeta})\operatorname{Ai}'\left(\zeta\right% )\operatorname{Bi}'\left(\overline{\zeta}\right),$
ⓘ Defines: $f_{n}$ : function (locally) Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\overline{\NVar{z}}$ : 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	$\displaystyle c_{0}(t)$	$\displaystyle=1,$
	$\displaystyle d_{0}(t)$	$\displaystyle=0,$
	$\displaystyle c_{n+1}(t)$	$\displaystyle=c_{n}^{\mspace{1.0mu}\prime}(t)+td_{n}(t),$
	$\displaystyle d_{n+1}(t)$	$\displaystyle=c_{n}(t)+d_{n}^{\mspace{1.0mu}\prime}(t).$
ⓘ 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