Gaussian积分 e(ikx)*e(-kx^2/2)积分区间无穷

take y=√kx
evaluate ∫e(i√ky)e(-y^2/2)dy/√k
f(y)=e(-y^2/2)
g(k)=∫e(i√ky)e(-y^2/2)dy /*g(k)=∫e(i√ky)f(y)dy*/
df(y)/dy+yf(y)=0
→∫e(i√ky)(df(y)/dy+yf(y)=0)dy
∫e(i√ky)df(y)/dydy=-i√kg(k) /*by parts*/
∫e(i√ky)yf(y)=-2i√kdg(k)/dk
→g(k)+2dg(k)/dk=0
→g(k)=Ce(-k/2)
where C=g(0)=∫e(-y^2/2)dy=√(2π) /*Gaussian*/
→g(k)=√(2π)e(-k/2)
→∫e(ikx)e(-kx^2/2)dx=g(k)/√k=√(2π/k)e(-k/2)
和计算器估算出来的结果相符
不出意外的话 应该是正解