What are the criteria for an elementary function to be infinitely integrable in elementary functions?

Question

What features of elementary functions define a class of functions whose consecutive indefinite integration also gives an elementary function?

Is there a way to check whether a given elementary function has such property?

Answer 1

Here's a potentially more practical criterion. It is based on the integration by parts identity $$ \int_0^x dy \int_0^y dz \, f(z) = x \int_0^x dz \, f(z) - \int_0^x dy \, y f(y) $$ If you don't like $0$ as a lower integration bound, pick another point in the domain of $f(x)$ or just absorb it into an overall additive constant.

Iterating the above identity gives the following conclusion: an elementary function $f(x)$ is $n$-times integrable in terms of elementary functions iff each of $x^k f(x)$, for $k=0,\ldots,n-1$, is once integrable in terms of elementary functions.

Answer 2

My recent papers

Chen, Shaoshi, Stability Problems in Symbolic Integration, MR4488863.

Chen, Shaoshi; Feng, Ruyong; Guo, Zewang; Lu, Wei, Stability problems on D-finite functions, ZBL07760759 MR4618424.

may partially answer this question. For a self-map $f$ from $A$ to $A$, an element $a$ in $A$ is said to be stable if there exists a sequence $a_i$ in $A$ such that $a_0=a$ and $a_i = f(a_{i+1})$ for all $i\geq 0$. Now taking $A$ to be the field of elementary functions and $f$ the derivation on $A$. This paper shows that $\exp(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f = ax + b$ with $a, b$ being constants in $\mathbb{C}$. Similarly, $\log(f(x))$ with $f$ being a rational-function in $\mathbb{C}(x)$ is stable if and only if $f$ is a Laurent polynomial in $\mathbb{C}[x, 1/x]$. In general, it is a challenging problem to characterize all possible stable elementary functions, especially the case of algebraic functions.

Answer 3

Each of the elementary functions of Liouville and Ritt is infinitely differentiable within the elementary functions. That is because the elementary functions form a differential field that is closed under differentiation.

The following are two examples of elementary functions with an n-th derivative which is predictable.

Let $m$, $n$ $\in\mathbb{N}_{+}$.

Your elementary functions shall be infinitely often integrable within the elementary functions. That means, all $n$-th degree antiderivatives of this functions have to be infinitely often differentiable within the elementary functions.

a)

A first simple class of such elementary functions are the polynomials. The antiderivative of a polynomial is again a polynomial. Therefore each $n$-th degree antiderivative of a polynomial is a polynomial again.

b)

A second simple class of such functions, $f$, are the elementary functions having one $n$-th degree derivative that is $f$ again:

$$\frac{d^{n}f(x)}{dx^{n}}=f(x)$$

Because the $n$-th degree antiderivative is an elementary function, all $m$-th degree antiderivatives with $m<n$ are elementary functions again. These differential equations can be easily solved.

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Also the Liouvillain functions form a differential field that is closed under differentiation. Therefore they are also infinitely differentiable in the Liouvillian functions. The elementary functions are a subfield of the Liouvillian functions.

Each function from a set which can be generated by successive integration from a self-differentiable function $x\mapsto ce^x$, $c$ a constant, is infinitely differentiable in this set.