Timeline for existence of antiderivatives of nasty but elementary functions
Current License: CC BY-SA 2.5
6 events
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| Mar 12, 2010 at 4:36 | comment | added | Jacques Carette | There is already a technical definition of "elementary function" that exists, and has existed since Liouville. The OP used it, rightly or wrongly. It's up to the OP to tell us if he erred in his usage or not. The answer to the question as you wrote it is simple: no, for trivial reasons -- it is trivial to write down everywhere undefined functions. If you restrict yourself to provably total functions, then the answer changes completely and becomes 'yes', but that is rather delicate to prove. | |
| Mar 12, 2010 at 4:33 | comment | added | David E Speyer | This is off topic (though interesting). The question is whether these functions have antiderivatives at all, not whether they have antiderivatives in closed form. | |
| Mar 12, 2010 at 4:06 | comment | added | Qiaochu Yuan | When you change the meaning of "elementary function" you are misinterpreting the OP's question. I think it should be interpreted as: "does a function you could 'write down' in a typical calculus course always have an antiderivative?" | |
| Mar 12, 2010 at 2:50 | history | edited | Gerald Edgar | CC BY-SA 2.5 |
added 1515 characters in body
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| Mar 11, 2010 at 23:21 | comment | added | Jacques Carette | You give a semantic definition of 'elementary', whereas ever since Liouville, the usual definition is syntactic -- see Bruce Westbury's answer for the 'common' definition. | |
| Mar 11, 2010 at 19:47 | history | answered | Gerald Edgar | CC BY-SA 2.5 |