Intereting Posts

Complex exponent $(z^{\alpha})^{\beta}$.
The sequence $a_n = \left(\frac{n}{n+1}\right)^{n+1}$ is increasing
Infinite Sequence of Inscribed Pentagrams – Where does it converge?
Fourier series of $\sqrt{1 – k^2 \sin^2{t}}$
Behaviour of $\int_0^{\frac{\pi^2}{4}}\exp(x\cos(\sqrt{t}))dt$
Modulus trick in programming
Use an induction argument to prove that for any natural number $n$, the interval $(n,n+1)$ does not contain any natural number.
If $a, b$ are relatively prime proof.
what happens to rank of matrices when singular matrix multiply by non-singular matrix??
Group theory applications along with a solved example
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
Quotient of polynomials, PID but not Euclidean domain?
Finding the location of an image of the Mandelbrot set
the representation of a free group
Galois group command for Magma online calculator?

It is known that there exist elementary functions which are not elementary integrable, i.e. there exists no elementary anti derivative. Example: $f(x) = e^{-x^2}$.

Let $A$ be the set of elementary functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Then:

- Add
*finite*many Riemann integrable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ to $A$. - Add all compositions of $A$-functions to $A$.
- Repeat 2. until the the set does not grow anymore. (Is this process guaranteed to end after finite steps? If not, 1. should only allow functions which leads to an end.)

Let $f \in A, g: \mathbb{R} \rightarrow \mathbb{R}, g(x) = \int_0^x f(s) \mathrm{d}s$. My question: Does there exist a set of functions (to be chosen in 1. so that $f \in A \Rightarrow g \in A$ is always true?

- Solving Some Transcendental Equations
- Injective map from real projective plane to $\Bbb{R}^4$
- Treatise on non-elementary integrable functions
- Algebraic numbers that cannot be expressed using integers and elementary functions
- Expressing the solutions of the equation $ \tan(x) = x $ in closed form.
- Prove that an equation has no elementary solution

PS: Sorry for the wording of the title, I failed to come up with something better.

**Edit:** If it makes the task easier, the functions in 1. may have a finite number of parameters. For example adding all the functions $f_k: x \mapsto \int_0^x e^{t^k} \mathrm{d}t$ is also allowed now.

- How to evaluate $ \int_0^1 {\log x \log(1-x) \log^2(1+x) \over x} \,dx $
- Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$
- Singularities of an integral
- Using trig substitution to evaluate $\int \frac{dt}{( t^2 + 9)^2}$
- Improper integrals and right-hand Riemann sums
- What IS the value of a complex line integral?
- How to evaluate$ \int_{\partial D} \int_{\partial D} \frac{1}{|x-y|}dxdy$, where $D$ is the sphere in 3D?
- Fundamental Theorem of Calculus Confusion regarding atan
- Integrating a product of exponentials and error functions
- Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$

Your process of composition cannot end after finite steps. Therefore your point 3 cannot be fulfilled.

Allow an infinite number of compositions in point 2 or restrict point 2 to a finite number of compositions instead.

To apply Liouville’s theory of integration in finite terms, the set $A$ has to be a differential field in each stage. That means you can add in your point 1 only integrals of functions from $A$.

In the answer to Why can’t we define more elementary functions?, it is shown that the process of adding necessary new transcendents (antiderivatives) is infinite in the case of Liouville’s conditions.

- Relationship among the function spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$
- What is the derivative of $x^i$?
- Local homeomorphisms which are not covering map?
- Prime factorization knowing n and Euler's function
- The Leibniz rule for the curl of the product of a scalar field and a vector field
- How to compute intersection multiplicity?
- Frechet differentiable implies reflexive?
- $\frac{d\Phi^{-1}(y)}{dy} = \frac{1}{\frac{d}{dy}}$?
- Is it possible to find an infinite set of points in the 3D space where the distance between any pair is rational?
- Infinite Series (Telescoping?)
- Sum of eigenvalues and singular values
- Construction of outer automorphisms of GL(n,K)
- Is $B = A^2 + A – 6I$ invertible when $A^2 + 2A = 3I$?
- Can you use both sides of an equation to prove equality?
- Transfinite Recursion Theorem