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Considering the complex logarithm, when do we have $$\exp\left(\sum_{i=1}^\infty a_i\right)=\prod_{i=1}^\infty \exp(a_i) ?$$

I originally wanted to try to prove it by showing $$\lim \prod^N \exp(a_i)=\exp\left(\lim \sum^N a_i\right),$$ but it doesn't seem to work.

Thank you.

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    $\begingroup$ Unfortunately, it is not clear what you are trying to ask, and your English doesn't seem to help. Are you asking when the first equality is true? $\endgroup$ – Alex M. Jul 28 '15 at 18:40
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    $\begingroup$ When the left side exists, the right side also exists, and the two are equal. It can happen that the left side doesn't exist and the right side does, for instance if $a_k = 2 k \pi i$. $\endgroup$ – Ian Jul 28 '15 at 18:42
  • $\begingroup$ @AlexM. Yes. Sorry if the post is messy. I will try to improve it. $\endgroup$ – k99731 Jul 28 '15 at 18:42
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I don't think this has anything to do with the complex logarithm function. If the series $\sum a_i$ converges, then the left-hand side will exist (because $\exp$ is continuous): $\exp (\sum \limits _{i=1} ^\infty a_i) = \exp (\lim \limits _{N \to \infty} \sum \limits _{i=1} ^N a_i) = \lim \limits _{N \to \infty} \exp (\sum \limits _{i=1} ^N a_i) = \lim \limits _{N \to \infty} \prod \limits _{i=1} ^N \exp (a_i) = \prod \limits _{i=1} ^\infty \exp (a_i)$.

If the series diverges, then the left-hand side may fail to exist, while the right-hand side may have a well-defined value (understood as the limit of a sequence of finite products), exactly as you have noticed in the first version of your question (before editing it).

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