Re: [微積] 瑕積分均勻收斂
※ 引述《tasukuchiyan (Tasuku)》之銘言：
: Does the improper integral
: ∞
: f(x) = ∫exp(-sx)sins/s ds
: 0
: converge uniformly on [0,∞)?
: 請問，有什麼方法可以證明它是否均勻收斂？謝謝。
(sins)/s , s ≠ 0
Let f(s,x) = e^(-sx), g(s,x) = { for all s,x in [0,∞).
1 , s = 0
Then f( ,x) is decreasing for x in [0,∞),
and ∥f∥≦ 1 for all x,s in [0,∞).
∞ ∞
Also, ∫(sins)/s ds converges ∴ ∫g( ,x) ds converges uniformly.
0 0
Hence, by Abel test we get this improper integral converging uniformly
on [0,∞).
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