# Thread: Validity of termwise integration

1. ## 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.

2. 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?

3. 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.

4. 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 ?

5. 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.

6. 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 $s\ge{0}$