燕尾積分

燕尾積分的分岔滿足下列方程式^[2]:p781

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

${\displaystyle y=-t(3z+10t^{2})}$ 

燕尾積分的斯托克斯曲線（Stokes curve）滿足下列方程式^[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${\displaystyle x_{+},x_{-}}$ 是下列五階代數方程的最小的兩個解：

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