:  ξ                       ∞
: ∫ exp(-x^2)dx = (1/√π)∫exp(-x^2/(2ξ)^2) [sin(x)/x] dx
:  0

賺個文章數

 ξ                1  +ξ                 1  +∞
∫ exp(-x^2) dx = ---∫   exp(-x^2) dx = ---∫   rect_ξ(x) exp(-x^2) dx
 0                 2  -ξ                 2  -∞

rect_ξ(x) = 1 if  -ξ=< x =<ξ

           = 0 else

   1    1
= --- ----- ∫∫∫dx dx' dk rect_ξ(x') exp-ik(x-x') exp(-x^2)
   2   2π

   1    1
= --- ----- ∫dk ∫dx exp(-x^2-ikx) ∫dx' rect_ξ(x') exp(ikx')
   2   2π

   1    1                             2 sin(kξ)
= --- ----- ∫dk ∫dx exp(-x^2-ikx) -------------
   2   2π                                k

  √π   +∞                  sin(kξ)
= ----- ∫   dk exp(-k^2/4) -----------
   2π   -∞                     k

    1     +∞                      sin(x)
= ----- ∫  dx exp(-x^2/(4ξ^2)) ---------
   √π   0                          x

※ 編輯: hellwize        來自: 36.227.147.62        (03/31 16:22)