作者josephw (Shower the People)
看板trans_math
标题Re: [积分] 一题积分
时间Mon Nov 19 14:18:47 2007
※ 引述《SCJhang (SCJhang)》之铭言:
: 请问如何证明下式:
: p-1
: ∞ x π
: ∫ ────dx = ────
: 0 1+x sinpπ
: 烦请板友指导,谢谢。
Suppose 0 < p < 1
x = t/(1-t) = 1/(1-t) - 1 => dx = 1/(1-t)^2
1 (t/(1-t))^(p-1) 1
=> ∫ ────────── . ──── dt
0 (1/(1-t)) (1-t)^2
1 (t)^(p-1)
= ∫ ────────── dt
0 (1-t)^p
1
= ∫ t^(p-1) (1-t)^(-p) dt
0
= B(p,-p) (by beta function)
= gamma(p) gamma(1-p) ( gamma(.) : gamma function)
= ....= π/sin(pπ)
最後一项卡关了....以前有证出来 抱歉 请高手再写好了
复变的方法不太行 也请高手写一下
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◆ From: 140.113.191.14
1F:推 dexter:sin函数可表为网页中(20)跟(21) 61.229.74.242 11/19 14:52
3F:→ dexter:而Gamma函数也可以表为infinite product 61.229.74.242 11/19 14:54
4F:→ dexter:这样就可以凑出来 61.229.74.242 11/19 15:00
5F:推 josephw:THX 终於记得了@@ 140.113.191.14 11/19 18:10
6F:推 SCJhang:josephw大,可以请你帮我完成它吗? 125.225.75.97 11/19 19:07
7F:→ SCJhang:凑不太出来 125.225.75.97 11/19 19:09
8F:推 GSXSP:本版2422 我有用复变算过 (p变成1/p)140.113.140.162 11/20 01:12
9F:→ GSXSP:讲错了 是p-1变成 -1/p' 1<p'<inf140.113.140.162 11/20 01:21
10F:推 SCJhang:GSXSP,感谢你的相助。 125.225.83.149 11/20 17:50