1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence of improper integrals with parameters

  1. Apr 17, 2012 #1
    I'm having a lot of trouble with the subject. Here's one example I'd like explained.
    [tex]F(t_1, t_2) = \int \limits_0^1 x^{t_1}\ln^{t_2}\frac{1}{x} dx[/tex]
    The book asks to find for what [itex]\vec{t}[/itex] F converges. The answer is [itex]\vec{t}\in(-1; \infty)^2[/itex], but I don't see how to get that.

    In general, what tools are there to test convergence? The only one I know is that (assuming that only f(b) is undefined) [itex]\int_a^b f[/itex] converges if [itex]\lim \limits_{x \rightarrow b} \int_x^b f = 0[/itex].
    Is it safe to assume that if [itex]g \sim f, x \rightarrow b[/itex] and [itex]\int g[/itex] converges then so does [itex]\int f[/itex]?

    My book shows some tests for uniform convergence, but that's not exactly the same thing, is it?
     
  2. jcsd
  3. Apr 17, 2012 #2
    use the change of variable u=ln(1/x), it becomes much clearer.
     
  4. Apr 18, 2012 #3
    So that's [itex]\int f = \int \limits_0^{\infty}e^{-(t_1+1)u}u^{t_2} \mathrm{d} u[/itex] then.
    It seems clear that [itex]\lim \limits_{u \rightarrow 0} f = u^{t_2}[/itex] and f converges at infinity when [itex]e^{-(t_1+1)u}[/itex] does.
    However, when t1 = 0, this integral is gamma function of t2+1. It also converges on negative non integers. Why did I not find that? Also, do there exist other values of t1 such that the integral converges for some t2<-1 ?
     
  5. Apr 18, 2012 #4
    According to this
    http://en.wikipedia.org/wiki/Gamma_function
    The integral representation of Gamma function is convergent only if t2+1>0. The full Gamma function is obtained by analytic continuation.
     
  6. Apr 19, 2012 #5
    Thanks, that's good to know.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Convergence of improper integrals with parameters
Loading...