查看“︁燕尾积分”︁的源代码

[[File:Swallowtail Integral Maple 3D plot.png|thumb|Swallowtail Integral Maple 3D plot]]
[[File:Swallowtail Integral Maple contour plot.png|thumb|Swallowtail Integral Maple contour plot]]
[[File:SwaIntegral Maple density plotllowtail.png|thumb|SwaIntegral Maple density plotllowtail]]
'''燕尾积分'''（Swallowtail Integral）是一种三阶多[[鞍点]][[积分]]，其定义如下<ref name=W>Roderick Wong, Asymptotic Approximation of Integrals,2001,SIAM.</ref><sup>:p388</sup>

<math>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.</math>

==分岔==
[[File:Swallowtail integral Bifurcation catastrophe.png|thumb|Swallowtail integral Bifurcation catastrophe]]
燕尾积分的分岔满足下列[[方程|方程式]]<ref name=F>Frank Oliver, NIST Handbook of Mathematical Functions, 2010,Cambridge University Press
</ref><sup>:p781</sup>

<math>x=3t^2(z++5t^2)</math>

<math>y=-t(3z+10t^2)</math>

==斯托克斯曲线==
[[File:Swallowtail integral Stokes set 1.png|thumb|150px|Swallowtail integral Stokes set 1]]
[[File:Swallowtail integral Stokes set 2.png|thumb|150px|Swallowtail integral Stokes set 2]]
[[File:Swallowtail integral catastrophe for z=0.png|thumb|150px|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<ref name=F>Frank Oliver, NIST Handbook of Mathematical Functions, 2010,Cambridge University Press
</ref><sup>:p783</sup>

<math>x=B_+|y|^{4/3}</math>

<math>x=B_{-}|y|^{4/3}</math>

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

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

其中<math>x_{+},x_{-}</math>是下列五阶[[代数方程]]的最小的两个解：

<math>80x^5-40x^4-55x^3+5x^2+20x-1=0</math>，即

<math>x_{+}=0.49730955169723075828e-1</math>

<math>x_{-}=0.74104357073646523281</math>

==相關==
*[[斯托克斯定理]]
*[[Euler–Maclaurin formula]]

==参考文献==
<references/>

[[Category:特殊函数]]