Could a computer theoretically compute all integrals in terms of some special functions or there need to be exist infinite number of such special functions to represent all integrals?

I know there theoretically could be exist a computer to prove all elementary geometry theorem.

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    $\begingroup$ I think so. We don't have closed form expressions for certain integrals, like $\int\frac{\sin(x)}{x}dx$, for example, but (at least to the best of my knowledge) all well-defined integrals are computable. $\endgroup$ – Newb Jan 31 '14 at 0:26
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    $\begingroup$ An uncountable set of integrals cannot be represented by a finite set of special functions. Perhaps, you meant all integrals that humans can directly consider? That would be a (theoretically) denumerable set. $\endgroup$ – Jacob Wakem Jan 31 '14 at 14:32

What is so special about special functions? They are functions that someone thought was special enough to give it a name. For example the gamma function is nothing special (at first glance). That being said, special functions are an "open set", so we can just define new special functions if needed and then any function can be represented as a special function.

$$\Xi (x;t) = \sum_{i=0}^t \int_0^x \Gamma(t)dt$$

I call this the Xi special function.....


For every real number r, there is an integrable function such that f(x)=r for all real x. Thus, there is an uncountable number of integrable functions. It follows directly that no computer can solve all integrals.


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