36 Integrals with Coalescing Saddles Properties 36.7 Zeros 36.9 Integral Identities

§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 in (36.2.1) in the exponent, expand the rest of the exponential as a power series in , 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 -dependent part of the exponential in powers of , 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 -dependent part of the exponential in powers of , 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

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

even,

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

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, : integer, : 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

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:: : open interval, : integer, : 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, : : factorial, : real parameter, : real parameter, : integer, : 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

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, : : factorial, $\realpart{}$ : real part, : real parameter, : real parameter, : integer, : 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, : 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:: : integer, : variable, $c_{n}(t)$ : coefficients and $d_{n}(t)$ : coefficients
Permalink:: http://dlmf.nist.gov/36.8.E6
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