皮尔西积分维基百科,自由的 encyclopedia 皮尔西积分(Pearcey Integral)是一种在论述光的传播、光的衍射、分岔理论、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[1] Integrand of Pearcey integral Pearcey Integral 3D Maple plot Pearcey Integral Maple contour plot Pearcey Integral Maple density plot 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)}}
皮尔西积分(Pearcey Integral)是一种在论述光的传播、光的衍射、分岔理论、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[1] Integrand of Pearcey integral Pearcey Integral 3D Maple plot Pearcey Integral Maple contour plot Pearcey Integral Maple density plot 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)}}