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

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

燕尾积分的分岔满足下列方程式^[2]:p781

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

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

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

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

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

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

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

其中${\displaystyle x_{+},x_{-}}$是下列五阶代数方程的最小的两个解：

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

${\displaystyle x_{+}=0.49730955169723075828e-1}$

${\displaystyle x_{-}=0.74104357073646523281}$

• 斯托克斯定理
• Euler–Maclaurin formula

1. ^ Roderick Wong, Asymptotic Approximation of Integrals,2001,SIAM.
2. ^ ^{2.0 2.1} Frank Oliver, NIST Handbook of Mathematical Functions, 2010,Cambridge University Press