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Math Help - Validity of termwise integration

  1. #1
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    Validity of termwise integration

    Hello everyone. I just joined the forum today and this is my first post. This looks like a great place with tons of active threads. I hope to contribute in the near future. I am fairly new to math and I hope to broaden my horizons here.

    But for now here is my question:

    When does the integral (from -∞ to ∞) of an infinite sum equal to the sum of the integral (from -∞ to ∞)? I am looking for a general theorem complete with proof. (Or does one even exist?)

    I have seen a few theorems dealing with definite integrals and infinite sums where you need uniform convergence to integrate termwise... but none that deal with infinite sums & integrals from from -∞ to ∞.



    If you know of an online source where I could find the theorem/proof could you post it for me please. Or I you have any insight feel free to share.
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  2. #2
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    To be a little more specific as to what I am looking at...it's basically:


    ∫ ∑s/[p(s-p)] where the sum is infinite (over p) and the limits of integration are from a-i∞ to a+i∞

    I have about 4 or 5 integral/sums like this.
    So instead of proving termwise integration is valid for each one...is there a more general approach?
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  3. #3
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    I assume that you mean

    \int_{-\infty}^\infty\left(\sum_p\frac{s}{p(s-p)}\right)ds.

    I know that for finite intervals you would be able to do term by term integration (M-test, so uniform convergence) but I'm not all too sure about the case for infinite intervals.
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  4. #4
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    In general, as you stated, uniform convergence is needed.

    Hmm for that.... what if you wrote \int_{-\infty}^\infty =\lim_{a\to -\infty}\lim_{b\to\infty} \int_a^b ?

    Or what if you can dominate the sum by a function whose integral over R is finite ?
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  5. #5
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    putnam120: I do mean to integrate from a - i∞ to a + i∞ i.e. integrating along a vertical line in C. Which is the limit as T approaches infinity of \int_{a-iT}^a+iT

    Moo: I believe I will have to do something like that. The sums that I am looking at all converge uniformly on any finite segment [a-ih , a+ih] a>1. *I still need to prove this*
    So termwise integration is valid on finite segments. I need to show that the limit of the sums as h approaches ∞ is equal to the sum of the limits. Then I can justify termwise integration (from a - i∞ to a + i∞ ) of the original sum.
    I found a useful lemma in Apostol's "Introduction to Analytic Number Theory" under grad text on p. 243 if anyone is interested. It gives some bounds on integrals such as the ones I would be dealing with. Since I don't know LaTEX (as you can see in the first sentence) it would be highly inefficient and most likely illegible if I were to try and write it out.

    Basically I was wondering if there was a more general theorem that deals with integrals and sums (both infinite). If so, I could just refer back to the theorem instead of proving termwise integration is valid for each specific integral/sum I have.

    Thanks for your responses. They are much appreciated.
    Last edited by The Shape; July 25th 2009 at 12:34 AM.
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  6. #6
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    Split it into

    \int\sum_{p=1}^{s-1}\frac{s}{p(s-p)}ds-\int\sum_{p=s+1}^\infty\frac{s}{p(p-s)}ds

    The first integral is finite so you can interchange the integral and the sum. As for the second define

    f_n(s)=\sum_{p=s+1}^{s+1+n}\frac{s}{p(p-s)}

    Then by the monotone convergence theorem we have

    \lim_{n\to\infty}\int f_n(s)ds=\int\lim_{n\to\infty}f_n(s)ds

    Which is exactly

    \sum\int f_n = \int\sum f_n.

    But this is assuming that s\ge{0}
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