Gaussian积分 e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,k k'参数

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Gaussian积分e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,kk''参数Gaussian积分e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,kk''参数Gaussian

Gaussian积分 e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,k k'参数
Gaussian积分 e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,k k'参数

Gaussian积分 e(ikx)*e(-k‘x^2/2)积分区间无穷,积分变量x,k k'参数
f(x)=e(-k′x^2/2)
g(k)=∫e(ikx)e(-k′x^2/2)dx /*g(k)=∫e(ikx)f(x)dx*/
df(x)/dx+k′xf(x)=0
→∫e(ikx)(df(x)/dx+k′xf(x)=0)dx
∫e(ikx)df(x)/dxdx=-ikg(k) /*by parts*/
∫e(ikx)k′xf(x)=-ik′dg(k)/dk
→kg(k)+k′dg(k)/dk=0
→g(k)=Ce(-k^2/(2k′))
where C=g(0)=∫e(-k′x^2/2)dx=√(2π/k′) /*Gaussian*/
→g(k)=√(2π/k′)e(-k^2/(2k′))
i.e.∫e(ikx)e(-kx^2/2)dx=g(k)=√(2π/k′)e(-k^2/(2k′))
FOURIER TRANSFORM OF GAUSSIAN FUNCTION