查看“︁皮尔西积分”︁的源代码

[[File:Pearcey integrand.gif|thumb|Integrand of Pearcey integral]]
[[File:Pearcey Integral 3D Maple plot.png|thumb|350px|Pearcey Integral 3D Maple plot]]
[[File:Pearcey Integral Maple contour plot.png|thumb|350px|Pearcey Integral  Maple contour plot]]
[[File:Pearcey Integral Maple density plot.png|thumb|350px|Pearcey Integral Maple density plot]]
'''皮尔西积分'''(Pearcey Integral)是一种在论述光的传播、[[光的衍射]]、[[分岔理論]]、[[突变理论]]以及关于[[特殊函数]]的渐进展开式的研究中常见的多鞍点积分，其定义为<ref name=P_p438>Paris,Hyperasymtotic Evaluation,p438</ref>

<math>P(x,y)=\int_{-\infty}^{\infty}\exp(I*(t^4+x*t^2+y*t))</math>

=<math>{-(1/2)*\sqrt(Pi)*exp(-(1/4*I)*y^2/x)*(limit(exp(I*t^4)*erf(I*x*t/\sqrt(-I*x)+(1/2*I)*y/\sqrt(-I*x)), t = infinity))/\sqrt(-I*x)+(1/2)*\sqrt(Pi)*exp(-(1/4*I)*y^2/x)*(limit(exp(I*t^4)*erf(I*x*t/\sqrt(-I*x)+(1/2*I)*y/\sqrt(-I*x)), t = -infinity))/\sqrt(-I*x)+(2*I)*\sqrt(Pi)*exp(-(1/4*I)*y^2/x)*(int(exp(I*t^4)*erf(I*x*t/\sqrt(-I*x)+(1/2*I)*y/\sqrt(-I*x))*t^3, t = -infinity .. infinity))/\sqrt(-I*x)}</math>

==被积分函数的鞍点==
对于大的<math>|x|</math>,皮尔逊积分的被积分函数左右各有三个鞍点<ref name=P_p438-9>Paris  p438-439</ref>。

在右平面的鞍点是

<math> rs1 := -(1/3)*\sqrt(6)*sinh((1/3)*arcsinh((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2)))           </math>

<math> rs2 := (1/3)*\sqrt(6)*sinh((1/3)*arcsinh((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2))+(1/3*I)*Pi)           </math>

<math> s3 := -(1/3*I)*\sqrt(6)*cosh((1/3)*arcsinh((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2))+(1/6*I)*Pi)           </math>

左平面的鞍点是：

<math>  ls1 := -(1/3)*\sqrt(6)*sin((1/3)*arcsin((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2))+(1/3)*Pi)               </math>

<math>  ls3 := (1/3)*\sqrt(6)*cos((1/3)*arcsin((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2))+(1/6)*Pi)               </math>

<math> ls2 := (1/3)*\sqrt(6)*sin((1/3)*arcsin((3/4)*y*\sqrt(2)*\sqrt(3)/x^(3/2)))                </math>

==分岔==
[[File:Bifurcation of Pearcey Integral.png|thumb|150px|Bifurcation of Pearcey Integral]]
皮尔西积分的分岔曲线为<ref name=F_p781>Frank Oliver, p781</ref>


<math>27*x^2=-8y^3</math>
==斯托克斯曲线==
[[File:Stokes set of Pearcey Integral.png|thumb|150px|Stokes set  of Pearcey Integral]]
皮尔西积分的斯托克斯曲线为<ref name=F_p783>Frank, p783</ref>

<math>y^3=\frac{27}{4}(\sqrt(27)-5)x^2</math>

==尖点突变==
[[File:Catastrophe Cusp of Pearcey Integral.png|thumb|200px|皮尔西积分的尖点突变区间]]
在（x,y)平面中，分岔曲线和斯托克斯曲线将平面分化为尖点突变区。<ref name=F_p784>Frank p784</ref>

==参考文献==
<references/>
*Frank Oliver, NIST Handbook of Mathematical Functions,2010,Cambridge University Press.
*R.B. Paris,D. Kaminski,Hyperasymptotic evaluation of the Pearcey integral via
Hadamard expansions.Journal of Computational and Applied Mathematics 190 (2006) 437–452

[[Category:特殊函数]]