皮尔西积分

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 P ( x , y ) = ∫ − ∞ ∞ exp ⁡ ( I ∗ ( t 4 + x ∗ t 2 + y ∗ t ) ) {\displaystyle P(x,y)=\int _{-\infty }^{\infty }\exp(I*(t^{4}+x*t^{2}+y*t))} = − ( 1 / 2 ) ∗ ( P i ) ∗ e x p ( − ( 1 / 4 ∗ I ) ∗ y 2 / x ) ∗ ( l i m i t ( e x p ( I ∗ t 4 ) ∗ e r f ( I ∗ x ∗ t / ( − I ∗ x ) + ( 1 / 2 ∗ I ) ∗ y / ( − I ∗ x ) ) , t = i n f i n i t y ) ) / ( − I ∗ x ) + ( 1 / 2 ) ∗ ( P i ) ∗ e x p ( − ( 1 / 4 ∗ I ) ∗ y 2 / x ) ∗ ( l i m i t ( e x p ( I ∗ t 4 ) ∗ e r f ( I ∗ x ∗ t / ( − I ∗ x ) + ( 1 / 2 ∗ I ) ∗ y / ( − I ∗ x ) ) , t = − i n f i n i t y ) ) / ( − I ∗ x ) + ( 2 ∗ I ) ∗ ( P i ) ∗ e x p ( − ( 1 / 4 ∗ I ) ∗ y 2 / x ) ∗ ( i n t ( e x p ( I ∗ t 4 ) ∗ e r f ( I ∗ x ∗ t / ( − I ∗ x ) + ( 1 / 2 ∗ I ) ∗ y / ( − I ∗ x ) ) ∗ t 3 , t = − i n f i n i t y . . i n f i n i t y ) ) / ( − I ∗ x ) {\displaystyle {-(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)}} 对于大的 | x | {\displaystyle |x|} ,皮尔逊积分的被积分函数左右各有三个鞍点。 在右平面的鞍点是 r s 1 := − ( 1 / 3 ) ∗ ( 6 ) ∗ s i n h ( ( 1 / 3 ) ∗ a r c s i n h ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) ) {\displaystyle rs1:=-(1/3)*{\sqrt {(}}6)*sinh((1/3)*arcsinh((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2)))} r s 2 := ( 1 / 3 ) ∗ ( 6 ) ∗ s i n h ( ( 1 / 3 ) ∗ a r c s i n h ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) + ( 1 / 3 ∗ I ) ∗ P i ) {\displaystyle rs2:=(1/3)*{\sqrt {(}}6)*sinh((1/3)*arcsinh((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/3*I)*Pi)} s 3 := − ( 1 / 3 ∗ I ) ∗ ( 6 ) ∗ c o s h ( ( 1 / 3 ) ∗ a r c s i n h ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) + ( 1 / 6 ∗ I ) ∗ P i ) {\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： l s 1 := − ( 1 / 3 ) ∗ ( 6 ) ∗ s i n ( ( 1 / 3 ) ∗ a r c s i n ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) + ( 1 / 3 ) ∗ P i ) {\displaystyle ls1:=-(1/3)*{\sqrt {(}}6)*sin((1/3)*arcsin((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/3)*Pi)} l s 3 := ( 1 / 3 ) ∗ ( 6 ) ∗ c o s ( ( 1 / 3 ) ∗ a r c s i n ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) + ( 1 / 6 ) ∗ P i ) {\displaystyle ls3:=(1/3)*{\sqrt {(}}6)*cos((1/3)*arcsin((3/4)*y*{\sqrt {(}}2)*{\sqrt {(}}3)/x^{(}3/2))+(1/6)*Pi)} l s 2 := ( 1 / 3 ) ∗ ( 6 ) ∗ s i n ( ( 1 / 3 ) ∗ a r c s i n ( ( 3 / 4 ) ∗ y ∗ ( 2 ) ∗ ( 3 ) / x ( 3 / 2 ) ) ) {\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 27 ∗ x 2 = − 8 y 3 {\displaystyle 27*x^{2}=-8y^{3}} 皮尔西积分的斯托克斯曲线为 y 3 = 27 4 ( ( 27 ) − 5 ) x 2 {\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。 Paris,Hyperasymtotic Evaluation,p438 Paris p438-439 Frank Oliver, p781 Frank, p783 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