作者CCWck (上吧上吧给我上吧)
看板Math
标题Re: [问题]积分 sinx / x...
时间Sat Feb 18 00:47:57 2006
sinx/x
=[sin(pi* x/pi) / pi* (x/pi)]
=sinc(x/pi)
取傅立叶转换对(可查表)
F{sinc(x/pi)}=pi*rect(pi*f)
取f=0处 可知
∞
∫ sinx/x dx =pi
-∞
因为sinc函数为偶函数 所以
∞
∫ sinx/x dx =pi/2
0
※ 引述《gary27 (小龟)》之铭言：
: 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
: 我的问题是...如果用这种方法,接下来要如何找出C就等於pi/2呢??
--
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◆ From: 140.113.141.196
1F：推 gary27:我把三种方法都收到微积分区^^ 02/18 01:20
2F：推 s0904213:这一招真赞 ^^ 02/18 22:14
3F：推 Frobenius :解法整理：(已变更,三种方法分别为下列前三篇) 07/05 14:42
4F：→ Frobenius :gary27 (#13zVs5tJ) (Math) (Laplace) 07/05 14:43
5F：→ Frobenius :akrsw (#13zVXsBw) (Math) (Fubini) 07/05 14:43
6F：→ Frobenius :CCWck (#13zVs5tJ) (Math) (Fourier) 07/05 14:43
8F：→ Frobenius :FATTY2108 (#10kCopAy) (trans_math) 07/05 14:44
9F：→ Frobenius :FATTY2108 (#12wZFSca) (trans_math) 07/05 14:44
11F：推 Frobenius :相关问题： 07/05 14:48
12F：→ Frobenius :yhliu (#13zVXsBw) (Math) 07/05 14:48
13F：→ Frobenius :obelisk0114 (#1CHS4AzW) (Math) 07/05 14:48
14F：→ Frobenius :vincentflame(#1Ch9BxA9) (Math) 07/05 14:48
15F：→ Frobenius :olivegad (#10fxap0s) (trans_math) 07/05 14:49
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17F：→ Frobenius :bonsaixxx (#1Cd30_Oj) (Math) 07/05 14:49
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