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$$\int^{+\infty}_{0}\sum ^{+\infty}_{n=1}e^{-\pi n^2t}t^{s/2-1}dt=\sum ^{+\infty}_{n=1}\int^{+\infty}_{0}e^{-\pi n^2t}t^{s/2-1}dt $$

I know the way to think is to realize that $e^{-\pi n^2t}$ is decaying really fast, but I have trouble showing it. What theorem should I use to prove the above equality?

Neil hawking
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jk001
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  • Uniform convergence then limits are interchangable. – Jake Freeman Jun 24 '21 at 19:14
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    Is monotone convergence theorem available? – Sangchul Lee Jun 24 '21 at 19:16
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    It's an immediate consequence of Tonelli's theorem [https://en.wikipedia.org/wiki/Fubini%27s_theorem#Tonelli's_theorem_for_non-negative_measurable_functions], because what you are summing/integrating is non-negative. – John Dawkins Jun 24 '21 at 19:17
  • @JohnDawkins what is being integrated need not be nonnegative since the OP never said what $s$ is. It could be a complex variable (with suitable real part). – KCd Jun 24 '21 at 19:19
  • Sorry, I was trying to figure out, and it is true that $s$ is real ($s>1$). This is part of the question in the equality between zeta, theta, and gamma function. – jk001 Jun 24 '21 at 19:21
  • Oh wow, I haven't had enough practice on the theorems in Lebesgue integral, but I think I got it. Thank you. – jk001 Jun 24 '21 at 19:26
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    I presumed $s>0$, else the integral on the right diverges and there's not much to say. If $s$ is merely real, the two sides are equal though maybe $=+\infty$. – John Dawkins Jun 24 '21 at 19:28
  • @jk001 To answer your question: Tonellis theorem or more generally Fubinis theorem since a sum is an integral over $\mathbb N$ with respect to the counting measure. See here for an answer on math.se. – vitamin d Jun 24 '21 at 19:39
  • @KCd: It does not matter what the sign of $s$ is, the integral (as a function of $n$ and $t$) is nonnegative (the product space here is $\mathbb{N}\times(0,\infty)$ with the corresponding Borel $\sigma$-algerbas. Fubini-Tonelli applies. In any event, as SangchulLee said, this can be dealt with Beppo-Levi's (monotone theorem) theorem – Mittens Jun 24 '21 at 20:53

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