Why we say this function have closed form while the other doesn't?
$\int\sin(x)\ dx = -\cos(x) + C$ have a closed form
while $\int\frac{\sin(x)}{x}\ dx = \textrm{Si}(x) + C$ does not have a closed form?
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2Depends on your definition of closed. See: http://math.stackexchange.com/questions/265780/how-to-determine-with-certainty-that-a-function-has-no-elementary-antiderivative – Aryabhata Mar 12 '14 at 01:07
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"No closed form" means "No expression that's as simple as I want it to be." – eyeballfrog Jul 21 '22 at 14:15
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The Si function is expressible as a hypergeometric function, another well known special function sometimes considered a closed form, so it is not “arbitrary”. – Тyma Gaidash Jul 21 '22 at 14:36
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"Closed form" isn't a generally accepted term. "In closed form" definitely means "in explicit finite terms". In addition to that, what a closed form is depends on the operations, functions and symbols you allow.
see Wikipedia: Closed-form expression
Some allow only elementary expressions to be closed-form expressions. Therefore $\cos(x)$ is in closed form but $\textrm{Si}(x)$ not.
The most general definition of closed-form expressions allow all expressions in named operations/functions. With that definition, both $\cos(x)$ and $\textrm{Si}(x)$ are in closed form.
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