Re: [問題]積分 sinx / x...

[CCWck (上吧上吧給我上吧)]

sinx/x
=[sin(pi* x/pi) / pi* (x/pi)]
=sinc(x/pi)

取傅立葉轉換對(可查表)
F{sinc(x/pi)}=pi*rect(pi*f)
取f=0處 可知
  ∞
∫  sinx/x dx  =pi
 -∞

因為sinc函數為偶函數 所以

 ∞
∫  sinx/x dx  =pi/2
 0


※ 引述《gary27 (小龜)》之銘言：
: 我知道用laplase
:  ∞
: ∫  sinx/x dx
:  0
:                 ∞
: L{sint/t}(s) = ∫ [sint/t * e^(-st) dt]
:                 0
: dL/ds = -∫[sint * e^(-st) dt
:       = -1/(1+s^2)
: L{sint/t}(s) =∫-1/(1+s^2) ds
:           = - arctan(s) + C
:  ∞
: ∫  sinx/x dx = L{sint/t}(0) = -arctan(0) + C
:  0
:                              = C
: 我的問題是...如果用這種方法,接下來要如何找出C就等於pi/2呢??

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推 gary27:我把三種方法都收到微積分區^^
推 s0904213:這一招真讚 ^^
推 Frobenius   :解法整理：(精華區已變更,三種方法分別為下列前三篇)
→ Frobenius   :gary27      (#13zVs5tJ) (Math)          (Laplace)
→ Frobenius   :akrsw       (#13zVXsBw) (Math)          (Fubini)
→ Frobenius   :CCWck       (#13zVs5tJ) (Math)          (Fourier)
→ Frobenius   :head109     (http://tinyurl.com/757sktu)(Residue)
→ Frobenius   :FATTY2108   (#10kCopAy) (trans_math)
→ Frobenius   :FATTY2108   (#12wZFSca) (trans_math)
→ Frobenius   :head109      http://tinyurl.com/757sktu (Residue)
推 Frobenius   :相關問題：
→ Frobenius   :yhliu       (#13zVXsBw) (Math)
→ Frobenius   :obelisk0114 (#1CHS4AzW) (Math)
→ Frobenius   :vincentflame(#1Ch9BxA9) (Math)
→ Frobenius   :olivegad    (#10fxap0s) (trans_math)
→ Frobenius   :dolphin0111 (#17k0tYE7) (trans_math)
→ Frobenius   :bonsaixxx   (#1Cd30_Oj) (Math)
→ Frobenius   :ntust661    (#1Cd58vZ4) (Math)
推 Frobenius   :Integrate[Sin[x]/x,x]:=http://tinyurl.com/264vld5
推 Frobenius   :RainIced    (#1Dh_LHlx) (Math)
→ Frobenius   :penny93730  (#1D-Y806T) (Math)
→ Frobenius   :JohnMash    (#1D-YSIKd) (Math)
→ Frobenius   :obelisk0114 (#1D-YshwE) (Math)
→ Frobenius   :justmeme    (#1FnOy4Js) (Math)
→ Frobenius   :JohnMash    (#1FnPA0lh) (Math)