皮爾西積分

皮爾西積分(Pearcey Integral)是一種在論述光的傳播、光的衍射、分岔理論、突變理論以及關於特殊函數的漸進展開式的研究中常見的多鞍點積分，其定義為^[1]

$P(x,y)=\int _{-\infty }^{\infty }\exp(I*(t^{4}+x*t^{2}+y*t))$

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

被積分函數的鞍點

對於大的 $|x|$ ,皮爾遜積分的被積分函數左右各有三個鞍點^[2]。

在右平面的鞍點是

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

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

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

左平面的鞍點是：

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

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

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

分岔

皮爾西積分的分岔曲線為^[3]

$27*x^{2}=-8y^{3}$

斯托克斯曲線

皮爾西積分的斯托克斯曲線為^[4]

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

尖點突變

在（x,y)平面中，分岔曲線和斯托克斯曲線將平面分化為尖點突變區。^[5]

參考文獻

[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

[P_p438-1] [1]
Paris,Hyperasymtotic Evaluation,p438

[P_p438-9-2] [2]
Paris p438-439

[F_p781-3] [3]
Frank Oliver, p781

[F_p783-4] [4]
Frank, p783

[F_p784-5] [5]
Frank p784

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[2]

[3]

[4]

[5]