高斯函数积分列表, 在这些表达式中, displaystyle, frac, sqrt, frac, 为标准正态概率密度函数, displaystyle, infty, frac, left, operatorname, left, frac, sqrt, right, right, 为对应的累积分布函数, 其中erf为误差函数, displaystyle, frac, 为欧文t函数, 英语, owen, function, 不定积分, 编辑, displaystyle, displaystyle, display. 在这些表达式中 ϕ x 1 2 p e 1 2 x 2 displaystyle phi x frac 1 sqrt 2 pi e frac 1 2 x 2 为标准正态概率密度函数 F x x ϕ t d t 1 2 1 erf x 2 displaystyle Phi x int infty x phi t dt frac 1 2 left 1 operatorname erf left frac x sqrt 2 right right 为对应的累积分布函数 其中erf为误差函数 T h a ϕ h 0 a ϕ h x 1 x 2 d x displaystyle T h a phi h int 0 a frac phi hx 1 x 2 dx 为欧文T函数 英语 Owen s T function 不定积分 编辑 ϕ x d x F x C displaystyle int phi x dx Phi x C x ϕ x d x ϕ x C displaystyle int x phi x dx phi x C x 2 ϕ x d x F x x ϕ x C displaystyle int x 2 phi x dx Phi x x phi x C x 2 k 1 ϕ x d x ϕ x j 0 k 2 k 2 j x 2 j C displaystyle int x 2k 1 phi x dx phi x sum j 0 k frac 2k 2j x 2j C x 2 k 2 ϕ x d x ϕ x j 0 k 2 k 1 2 j 1 x 2 j 1 2 k 1 F x C displaystyle int x 2k 2 phi x dx phi x sum j 0 k frac 2k 1 2j 1 x 2j 1 2k 1 Phi x C 在这些积分中 n 为双阶乘 若n为偶数 则等于从2到n的所有偶数的乘积 若n为奇数 则等于从1到n的所有奇数的乘积 特别地 0 1 1 ϕ x 2 d x 1 2 p F x 2 C displaystyle int phi x 2 dx frac 1 2 sqrt pi Phi left x sqrt 2 right C ϕ x ϕ a b x d x 1 t ϕ a t F t x a b t C t 1 b 2 displaystyle int phi x phi a bx dx frac 1 t phi left frac a t right Phi left tx frac ab t right C qquad t sqrt 1 b 2 x ϕ a b x d x 1 b 2 ϕ a b x a F a b x C displaystyle int x phi a bx dx frac 1 b 2 left phi a bx a Phi a bx right C x 2 ϕ a b x d x 1 b 3 a 2 1 F a b x a b x ϕ a b x C displaystyle int x 2 phi a bx dx frac 1 b 3 left a 2 1 Phi a bx a bx phi a bx right C ϕ a b x n d x 1 b n 2 p n 1 F n a b x C displaystyle int phi a bx n dx frac 1 b sqrt n 2 pi n 1 Phi left sqrt n a bx right C F a b x d x 1 b a b x F a b x ϕ a b x C displaystyle int Phi a bx dx frac 1 b left a bx Phi a bx phi a bx right C x F a b x d x 1 2 b 2 b 2 x 2 a 2 1 F a b x b x a ϕ a b x C displaystyle int x Phi a bx dx frac 1 2b 2 left b 2 x 2 a 2 1 Phi a bx bx a phi a bx right C x 2 F a b x d x 1 3 b 3 b 3 x 3 a 3 3 a F a b x b 2 x 2 a b x a 2 2 ϕ a b x C displaystyle int x 2 Phi a bx dx frac 1 3b 3 left b 3 x 3 a 3 3a Phi a bx b 2 x 2 abx a 2 2 phi a bx right C x n F x d x 1 n 1 x n 1 n x n 1 F x x n ϕ x n n 1 x n 2 F x d x C displaystyle int x n Phi x dx frac 1 n 1 left left x n 1 nx n 1 right Phi x x n phi x n n 1 int x n 2 Phi x dx right C x ϕ x F a b x d x b t ϕ a t F x t a b t ϕ x F a b x C t 1 b 2 displaystyle int x phi x Phi a bx dx frac b t phi left frac a t right Phi left xt frac ab t right phi x Phi a bx C qquad t sqrt 1 b 2 F x 2 d x x F x 2 2 F x ϕ x 1 p F x 2 C displaystyle int Phi x 2 dx x Phi x 2 2 Phi x phi x frac 1 sqrt pi Phi left x sqrt 2 right C e c x ϕ b x n d x e c 2 2 n b 2 b n 2 p n 1 F b 2 x n c b n C b 0 n gt 0 displaystyle int e cx phi bx n dx frac e frac c 2 2nb 2 b sqrt n 2 pi n 1 Phi left frac b 2 xn c b sqrt n right C qquad b neq 0 n gt 0 定积分 编辑 x 2 ϕ x n d x 1 n 3 2 p n 1 displaystyle int infty infty x 2 phi x n dx frac 1 sqrt n 3 2 pi n 1 0 ϕ a x F b x d x 1 2 p a p 2 arctan b a displaystyle int infty 0 phi ax Phi bx dx frac 1 2 pi a left frac pi 2 arctan left frac b a right right 0 ϕ a x F b x d x 1 2 p a p 2 arctan b a displaystyle int 0 infty phi ax Phi bx dx frac 1 2 pi a left frac pi 2 arctan left frac b a right right 0 x ϕ x F b x d x 1 2 2 p 1 b 1 b 2 displaystyle int 0 infty x phi x Phi bx dx frac 1 2 sqrt 2 pi left 1 frac b sqrt 1 b 2 right 0 x 2 ϕ x F b x d x 1 4 1 2 p b 1 b 2 arctan b displaystyle int 0 infty x 2 phi x Phi bx dx frac 1 4 frac 1 2 pi left frac b 1 b 2 arctan b right 0 x ϕ x 2 F x d x 1 4 p 3 displaystyle int 0 infty x phi x 2 Phi x dx frac 1 4 pi sqrt 3 0 F b x 2 ϕ x d x 1 2 p arctan b arctan 1 2 b 2 displaystyle int 0 infty Phi bx 2 phi x dx frac 1 2 pi left arctan b arctan sqrt 1 2b 2 right F a b x 2 ϕ x d x F a 1 b 2 2 T a 1 b 2 1 1 2 b 2 displaystyle int infty infty Phi a bx 2 phi x dx Phi left frac a sqrt 1 b 2 right 2T left frac a sqrt 1 b 2 frac 1 sqrt 1 2b 2 right x F a b x 2 ϕ x d x 2 b 1 b 2 ϕ a t F a 1 b 2 1 2 b 2 displaystyle int infty infty x Phi a bx 2 phi x dx frac 2b sqrt 1 b 2 phi left frac a t right Phi left frac a sqrt 1 b 2 sqrt 1 2b 2 right F b x 2 ϕ x d x 1 p arctan 1 2 b 2 displaystyle int infty infty Phi bx 2 phi x dx frac 1 pi arctan sqrt 1 2b 2 x ϕ x F b x d x x ϕ x F b x 2 d x b 2 p 1 b 2 displaystyle int infty infty x phi x Phi bx dx int infty infty x phi x Phi bx 2 dx frac b sqrt 2 pi 1 b 2 F a b x ϕ x d x F a 1 b 2 displaystyle int infty infty Phi a bx phi x dx Phi left frac a sqrt 1 b 2 right x F a b x ϕ x d x b t ϕ a t t 1 b 2 displaystyle int infty infty x Phi a bx phi x dx frac b t phi left frac a t right qquad t sqrt 1 b 2 0 x F a b x ϕ x d x b t ϕ a t F a b t 1 2 p F a t 1 b 2 displaystyle int 0 infty x Phi a bx phi x dx frac b t phi left frac a t right Phi left frac ab t right frac 1 sqrt 2 pi Phi a qquad t sqrt 1 b 2 ln x 2 1 s ϕ x s d x ln s 2 g ln 2 ln s 2 1 27036 displaystyle int infty infty ln x 2 frac 1 sigma phi left frac x sigma right dx ln sigma 2 gamma ln 2 approx ln sigma 2 1 27036 取自 https zh wikipedia org w index php title 高斯函数积分列表 amp oldid 54759682, 维基百科,wiki,书籍,书籍,图书馆,