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3
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2answers
85 views

How does one make real functions a differentiable field?

If you want to apply the results of differential field theory to actual $\Bbb R\to\Bbb R$ functions, then first of all you have to find operations that make these functions a field. The trouble is ...
65
votes
1answer
4k views

How to determine with certainty that a function has no elementary antiderivative?

Given an expression such as $f(x) = x^x$, is it possible to provide a thorough and rigorous proof that there is no function $F(x)$ (expressible in terms of known algebraic and transcendental ...
0
votes
2answers
406 views

$\int \cos(x) \ln(x) dx$, elementary function?

My course book bluntly mentions (freely translation without any proof): Integral functions with the terms $x^{\alpha} \sin(\beta x)$, $x^{\alpha} \cos(\beta x)$ or $x^{\alpha}e^{\beta x}$ ...
1
vote
1answer
453 views

Is a factorial an algebraic function and an elementary function?

Following is a question spun off from a comment I received: is a factorial an elementary function and an algebraic function? From elementary functions by Wikipedia By starting with the ...