Need a hint proving that integral converges

  • Thread starter Tricore
  • Start date
  • #1
21
0
I am trying to prove that [tex]\int_0^{\pi/4} (\tan x)^{-\alpha} dx[/tex] is finite for [tex]0 < \alpha < 1[/tex], where the integral denotes the Lebesgue integral with the Lebesgue measure. I've decided wether it converges for all other values of [tex]\alpha\in\mathbb{R}[/tex], but am simply stuck with this one.

I've tried using monotone convergence to integrate over the closed interval from 1/n to pi/4, where I can use the Riemann integral, but I can't figure out how to find an anti-derivative for the function. I've also tried finding a larger, integrable function but without luck.

Any hints would be much appreciated.
 

Answers and Replies

  • #2
22,129
3,297
What happens if [tex]\alpha=1[/tex]. Is that function integrable?
 
  • #3
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,950
19
The only difficulty in this problem is the poles that the integrand has at 0 and pi/4, right? So find another function that has the same poles, and study that function, and the difference between it and your integrand.
 

Related Threads on Need a hint proving that integral converges

  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
21
Views
1K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
16
Views
2K
  • Last Post
Replies
5
Views
980
  • Last Post
Replies
1
Views
988
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
11
Views
693
Top