皮尔西积分

对于大的${\displaystyle |x|}$ ,皮尔逊积分的被积分函数左右各有三个鞍点^[2]。

在右平面的鞍点是

${\displaystyle rs1:=-(1/3)*{\sqrt {(}}6)*sinh((1/3)*arcsinh((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2)))}$ 

${\displaystyle rs2:=(1/3)*{\sqrt {(}}6)*sinh((1/3)*arcsinh((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/3*I)*Pi)}$ 

${\displaystyle s3:=-(1/3*I)*{\sqrt {(}}6)*cosh((1/3)*arcsinh((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/6*I)*Pi)}$ 

左平面的鞍点是：

${\displaystyle ls1:=-(1/3)*{\sqrt {(}}6)*sin((1/3)*arcsin((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/3)*Pi)}$ 

${\displaystyle ls3:=(1/3)*{\sqrt {(}}6)*cos((1/3)*arcsin((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/6)*Pi)}$ 

${\displaystyle ls2:=(1/3)*{\sqrt {(}}6)*sin((1/3)*arcsin((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2)))}$ 

皮尔西积分的分岔曲线为^[3]

${\displaystyle 27*x^{2}=-8y^{3}}$ 

皮尔西积分的斯托克斯曲线为^[4]

${\displaystyle y^{3}={\frac {27}{4}}({\sqrt {(}}27)-5)x^{2}}$ 

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。^[5]

1. ^ Paris,Hyperasymtotic Evaluation,p438
2. ^ Paris p438-439
3. ^ Frank Oliver, p781
4. ^ Frank, p783
5. ^ Frank p784

• 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