燕尾積分(Swallowtail Integral)是一種三階多鞍點積分,其定義如下[1]:p388 Swallowtail Integral Maple 3D plot Swallowtail Integral Maple contour plot SwaIntegral Maple density plotllowtail P ( x 1 , x 2 , x 3 ) = ∫ t = − ∞ ∞ e x p ( I ∗ ( t 5 + x [ 1 ] ∗ t + x [ 2 ] ∗ t 2 + x [ 3 ] ∗ t 3 ) ) d t . {\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.} Swallowtail integral Bifurcation catastrophe 燕尾積分的分岔滿足下列方程式[2]:p781 x = 3 t 2 ( z + + 5 t 2 ) {\displaystyle x=3t^{2}(z++5t^{2})} y = − t ( 3 z + 10 t 2 ) {\displaystyle y=-t(3z+10t^{2})} Swallowtail integral Stokes set 1 Swallowtail integral Stokes set 2 Swallowtail integral catastrophe for z=0 燕尾積分的斯托克斯曲線(Stokes curve)滿足下列方程式[2]:p783 x = B + | y | 4 / 3 {\displaystyle x=B_{+}|y|^{4/3}} x = B − | y | 4 / 3 {\displaystyle x=B_{-}|y|^{4/3}} B + = 10 − 1 / 3 ( 2 x + 4 / 3 − 1 2 ( x + − 2 / 3 ) {\displaystyle B_{+}=10^{-1/3}(2x_{+}^{4/3}-{\frac {1}{2}}(x_{+}^{-2/3})} B − = 10 − 1 / 3 ( 2 x − 4 / 3 − 1 2 ( x − − 2 / 3 ) {\displaystyle B_{-}=10^{-1/3}(2x_{-}^{4/3}-{\frac {1}{2}}(x_{-}^{-2/3})} 其中 x + , x − {\displaystyle x_{+},x_{-}} 是下列五階代數方程的最小的兩個解: 80 x 5 − 40 x 4 − 55 x 3 + 5 x 2 + 20 x − 1 = 0 {\displaystyle 80x^{5}-40x^{4}-55x^{3}+5x^{2}+20x-1=0} ,即 x + = 0.49730955169723075828 e − 1 {\displaystyle x_{+}=0.49730955169723075828e-1} x − = 0.74104357073646523281 {\displaystyle x_{-}=0.74104357073646523281} 斯托克斯定理 Euler–Maclaurin formula [1]Roderick Wong, Asymptotic Approximation of Integrals,2001,SIAM. [2]Frank Oliver, NIST Handbook of Mathematical Functions, 2010,Cambridge University Press Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for Firefox
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