Is there a class of functions closed against differentiation besides elementary? [closed]

Is there a finite set $P$ of non-elementary functions $f_n$ such that the derivative of any function $f$ from that set is not elementary, but expressible with functions from the same set $P$ plus elementary functions?

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closed as unclear what you're asking by Misha, Ricardo Andrade, Stefan Kohl, Andrey Rekalo, Chris GodsilMar 17 '14 at 11:49

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What do you mean by "expressible"? Compositions and algebraic operations? Partial inverses of the functions from your class (e.g. radicals)? – Misha Mar 16 '14 at 14:09
@Misha built from finite number of functions from P and elementary functions using arithmetic operations and composition. – Anixx Mar 16 '14 at 14:13
OK, what functions do you regard as "elementary" (there is no consistent terminology here). Functions from where to where? (Real or complex.) For instance, would $P=\{erf\}$ satisfy you? Please, think through what you are really asking and update your question. – Misha Mar 16 '14 at 14:16
If I'm understanding the question right, let $g(x)$ an elementary function such that $f(x) = \int g(x)\ dx$ is not elementary. Then $P = \{f(x)\}$ works. – rghthndsd Mar 16 '14 at 14:20
@Misha This is not my understanding. When asked, Anixx linked to the wikipedia article which states such functions can be complex. Maybe I'm confused. – rghthndsd Mar 16 '14 at 14:26

Let $g$ be an elementary function whose indefinite integral $f$ is nonelementary, and set $P = \{\cos(t)f(t), \sin(t)f(t)\}$.
• The Bessel functions, $\{J_\nu(x), K_\nu(x) | \nu\in\mathbb{Z}\}$
• Airy functions, $\{Ai(x), Ai^\prime(x)\}$, $\{Bi(x), Bi^\prime(x)\}$
• Complete elliptic integrals, $\{E(x), K(x)\}$
And also many other solution sets to classes of differential equations. You can also replace $x$ by any elementary function in these classes.
@Anixx For the Bessel functions, you can take any two consecutive indices, since they have recurrence identities. I.e., you can express the derivative of $I_\nu$ and $K_\nu$ in terms of $I_{\nu+1},K_{\nu+1}$ or $I_{\nu-1},K_{\nu-1}$ (in addition to $I_\nu,K_\nu$ themselves). That was unclear in my answer, sorry. The other two or three examples are already finite. – Christopher Creutzig Mar 17 '14 at 6:52