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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1$\begingroup$ What do you mean by "expressible"? Compositions and algebraic operations? Partial inverses of the functions from your class (e.g. radicals)? $\endgroup$– MishaCommented Mar 16, 2014 at 14:09
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$\begingroup$ @Misha built from finite number of functions from P and elementary functions using arithmetic operations and composition. $\endgroup$– AnixxCommented Mar 16, 2014 at 14:13
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1$\begingroup$ 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. $\endgroup$– MishaCommented Mar 16, 2014 at 14:16
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1$\begingroup$ 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. $\endgroup$– rghthndsdCommented Mar 16, 2014 at 14:20
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1$\begingroup$ @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. $\endgroup$– rghthndsdCommented Mar 16, 2014 at 14:26
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2 Answers
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Let $g$ be an elementary function whose indefinite integral $f$ is nonelementary, and set $P = \{\cos(t)f(t), \sin(t)f(t)\}$.
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- 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.
answered Mar 16, 2014 at 18:52
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$\begingroup$ Are these classes have finite number of functions? It seems the derivatives of Bessel functions give Bessel functions with another index, and the derivative by index is not elementary. Complete elliptic integrals indeed seem to fit. $\endgroup$– AnixxCommented Mar 17, 2014 at 0:04
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$\begingroup$ @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. $\endgroup$ Commented Mar 17, 2014 at 6:52