# Analytic function whose derivatives and primitives are independent from a given set of countable cardinality

Let $$L=(l_j)_{j\in\mathbb{N}}$$ be a set of countably many independent real analytic functions on $$[0,2\pi]$$. Here and in the following, independent means that a function cannot be written as finite linear combination of the others.

Can I always find an analytic function on $$[0,2\pi]$$ such that the function itself, all its derivatives and its primitives (computed as indefinite integral $$\int_0^x$$) are independent from L? How do I construct it?

For example, if $$L=(l_j):=(x^j)_{j\in\mathbb{N}}$$, then $$e^x$$, all its derivatives and all its primitives (in this case $$e^x$$ as well) are independent from L, since $$e^x$$ is not a polynomial. How do I approach the problem in general? Thanks.

• No matter what $L$ is, $e^{ax}$ will work for some $a$. – Wojowu Feb 12 at 10:09
• How does one see that in a rigorous way? – Leonardo Feb 12 at 11:20
• We may assume $L$ contains $x^j$ for all $j$. If $e^{ax}$ or any of its integrals or derivatives was linearly dependent on $L$, then $e^{ax}$ would belong to the linear span of $L$. But the latter has countable dimension, while $e^{ax}$ form an uncountable linearly independent set. – Wojowu Feb 12 at 11:36