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Closed forms: What they are and why they matter

The term "closed form" is one of those mathematical notions that is commonplace, yet virtually devoid of rigor. And, there is disagreement even on the intuitive side; for example, most everyone would say that (\pi + log 2) is a 
closed form, but some of us would think that the Euler constant $\gamma$ is not closed. Like others before us, we shall try to supply some missing rigor to the notion of 
closed forms and also to give examples from modern research where the question of closure looms both important and elusive.