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Homework Help: 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
    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.
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