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.