§36.8 Convergent Series Expansions

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 $\mathop{\mathrm{Ai}\/}\nolimits$ and ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}$ . 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).
Keywords:: Pearcey integral, canonical integrals, elliptic umbilic canonical integral, hyperbolic umbilic canonical integral, swallowtail canonical integral
Referenced by:: §36.15(i)
Permalink:: http://dlmf.nist.gov/36.8

36.8.1

$\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)=\dfrac{2}{K+2}\sum\limits _{{n=0}}^{\infty}\mathop{\exp\/}\nolimits\!\left(i\dfrac{\pi(2n+1)}{2(K+2)}\right)\mathop{\Gamma\/}\nolimits\!\left(\dfrac{2n+1}{K+2}\right)a_{{2n}}(\mathbf{x}),$ $K$ even,

$\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)=\dfrac{2}{K+2}\sum\limits _{{n=0}}^{\infty}i^{n}\mathop{\cos\/}\nolimits\!\left(\dfrac{\pi(n(K+1)-1)}{2(K+2)}\right)\mathop{\Gamma\/}\nolimits\!\left(\dfrac{n+1}{K+2}\right)a_{n}(\mathbf{x}),$ $K$ odd,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\exp\/}\nolimits z$ : exponential function, $m$ : integer, $K$ : codimension and $a_{n}(\mathbf{x})$ : coefficients
Referenced by:: §36.8
Permalink:: http://dlmf.nist.gov/36.8.E1
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where

36.8.2

$a_{0}(\mathbf{x})=1,$

$a_{{n+1}}(\mathbf{x})=\dfrac{i}{n+1}\sum _{{p=0}}^{{\min(n,K-1)}}(p+1)x_{{p+1}}a_{{n-p}}(\mathbf{x}),$ $n=0,1,2,\dots$ .

Symbols:: $(a,b)$ : open interval, $m$ : integer, $K$ : codimension, $x_{i}$ : real parameter and $a_{n}(\mathbf{x})$ : coefficients
Permalink:: http://dlmf.nist.gov/36.8.E2
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For multinomial power series for $\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$ , see Connor and Curtis (1982).

36.8.3 $\dfrac{3^{{2/3}}}{4\pi^{2}}\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(3^{{1/3}}\mathbf{x}\right)=\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(y\right)\sum\limits _{{n=0}}^{\infty}(-3^{{-1/3}}iz)^{n}\dfrac{c_{n}(x)c_{n}(y)}{n!}+\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right){\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(y\right)\sum\limits _{{n=2}}^{\infty}(-3^{{-1/3}}iz)^{n}\dfrac{c_{n}(x)d_{n}(y)}{n!}+{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)\mathop{\mathrm{Ai}\/}\nolimits\!\left(y\right)\sum\limits _{{n=2}}^{\infty}(-3^{{-1/3}}iz)^{n}\dfrac{d_{n}(x)c_{n}(y)}{n!}+{\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right){\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(y\right)\sum\limits _{{n=1}}^{\infty}(-3^{{-1/3}}iz)^{n}\dfrac{d_{n}(x)d_{n}(y)}{n!},$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $!$ : $n!$ : factorial, $y$ : 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
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and

36.8.4 $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)=2\pi^{2}\left(\dfrac{2}{3}\right)^{{2/3}}\sum\limits _{{n=0}}^{\infty}\dfrac{\left(-i(2/3)^{{2/3}}z\right)^{n}}{n!}\realpart{\left(f_{n}\left(\dfrac{x+iy}{12^{{1/3}}},\dfrac{x-iy}{12^{{1/3}}}\right)\right)},$

Symbols:: $\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$ : canonical integral, $!$ : $n!$ : factorial, $\realpart{}$ : 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

where

36.8.5 $f_{n}(\zeta,\zeta^{{\ast}})=c_{n}(\zeta)c_{n}(\zeta^{{\ast}})\mathop{\mathrm{Ai}\/}\nolimits\!\left(\zeta\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\zeta^{{\ast}}\right)+c_{n}(\zeta)d_{n}(\zeta^{{\ast}})\mathop{\mathrm{Ai}\/}\nolimits\!\left(\zeta\right){\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(\zeta^{{\ast}}\right)+d_{n}(\zeta)c_{n}(\zeta^{{\ast}}){\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\zeta\right)\mathop{\mathrm{Bi}\/}\nolimits\!\left(\zeta^{{\ast}}\right)+d_{n}(\zeta)d_{n}(\zeta^{{\ast}}){\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\zeta\right){\mathop{\mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(\zeta^{{\ast}}\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $m$ : integer, $f_{n}$ : function, $c_{n}(t)$ : coefficients and $d_{n}(t)$ : coefficients
Permalink:: http://dlmf.nist.gov/36.8.E5
Encodings:: TeX, pMML, png

with asterisks denoting complex conjugates, and

36.8.6

$c_{0}(t)=1,$

$d_{0}(t)=0,$

$c_{{n+1}}(t)=c_{n}^{{\mspace{1.0mu}\prime}}(t)+td_{n}(t),$

$d_{{n+1}}(t)=c_{n}(t)+d_{n}^{{\mspace{1.0mu}\prime}}(t).$

Symbols:: $m$ : integer, $t$ : variable, $c_{n}(t)$ : coefficients and $d_{n}(t)$ : coefficients
Permalink:: http://dlmf.nist.gov/36.8.E6
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