燕尾积分

燕尾积分（Swallowtail Integral）是一种三阶多鞍点积分，其定义如下^[1]^:p388

Swallowtail Integral Maple 3D plot

Swallowtail Integral Maple contour plot

SwaIntegral Maple density plotllowtail

$P(x_{1},x_{2},x_{3})=\int _{t=-\infty }^{\infty }exp(I*(t^{5}+x[1]*t+x[2]*t^{2}+x[3]*t^{3}))\,dt.$

分岔

Swallowtail integral Bifurcation catastrophe

燕尾积分的分岔满足下列方程式^[2]^:p781

$x=3t^{2}(z++5t^{2})$

$y=-t(3z+10t^{2})$

Swallowtail integral Stokes set 1

Swallowtail integral Stokes set 2

Swallowtail integral catastrophe for z=0

燕尾积分的斯托克斯曲线（Stokes curve）满足下列方程式^[2]^:p783

$x=B_{+}|y|^{4/3}$

$x=B_{-}|y|^{4/3}$

$B_{+}=10^{-1/3}(2x_{+}^{4/3}-{\frac {1}{2}}(x_{+}^{-2/3})$

$B_{-}=10^{-1/3}(2x_{-}^{4/3}-{\frac {1}{2}}(x_{-}^{-2/3})$

其中 $x_{+},x_{-}$ 是下列五阶代数方程的最小的两个解：

$80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0$ ，即

$x_{+}=0.49730955169723075828e-1$

$x_{-}=0.74104357073646523281$

^ Roderick Wong, Asymptotic Approximation of Integrals,2001,SIAM.
^ ^2.0 ^2.1 Frank Oliver, NIST Handbook of Mathematical Functions, 2010,Cambridge University Press