At the end of my article Hyperelliptic action integral, Annales de l'institut Fourier 49(1), p.303–331, I state the following conjecture:

Conjecture about a singularity
Let D be the open unit disk in the complex plane and U_1,U_2,\,\dots\,,U_n be a cover of the punctured disk D*= D\{0} by open connected sets. Suppose on each open set U_j there is an injective holomorphic function f_j such that df_j=df_k on every intersection U_j\cap U_k. Then those differentials glue together to a meromorphic 1-form on D.

It is evident that the 1-form is holomorphic on D*. In the case that its residue at the origin vanishes, the conjecture follows because the singularity can be at most a pole, as can easily be derived from Picard's big theorem (cited below). But I am unable to prove the conjecture for the case of a non-zero residue.
I would be grateful for any proof or counter-example — well, trule spoken, a counter-example would be less welcome since I hope that the conjecture is true. I am guided by my simple geometric intuition on Riemann surfaces...

In 1880 Charles Emile Picard (1856-1941) proved the following theorem.

Picard's Big Theorem
A holomorphic function takes in any neighborhood of an essential singularity every complex value, with at most one exception, infinitely often.

A typical example for Picard's theorem

The function defined by

is holomorphic on \mathbb{C}\backslash0 and has an essential singularity in 0. Is there a value that is does not take (Picard says "at most one exception")? Yes, since f(z)\neq0 for all z\in\mathbb{C}\backslash0, there is an exceptional value and it must be zero; according to Picard's theorem there are, for any complex value w\neq0 and any \epsilon>0, infinitely many complex numbers z such that 0<|z|<\epsilon and f(z)=w.

Direct compuation of this example

In fact, for the example above the theorem of Picard is not really necessary, because we can also see what happens by direct computation: let w be a non-zero complex value and let \epsilon>0. There existe two real numbers r>0 and \varphi such that

For n \in \mathbb{N} let u_n=\ln r+i(\varphi+2\pi n) and z_n=1/{u_n}. Thus we have \lim_{n\to\infty}z_n=0.
f(z_n)=e^{u_n}=e^{\ln r+i(\varphi+2\pi n)}=re^{i \varphi}=w.

By taking n big enough we see that w has infinitely many pre-images in the punctured disk 0<\,|z|\,<\epsilon\,.

A less simple example

Denote by P the set of all prime numbers and consider the following function
g(z)=\sum_{p \in P}^{}\frac{1}{p!z^p}\;.

Since there is an essential singularity we can apply Picard's Theorem.
But a direct computation seems impossible...