Request for hint: introductory complex analysis problem _This is not a ‘do-my-homework’ question._ The problem is > For what value of $a$, is $$F(z)=\int^z_{z_0}e^z\left(\frac1z+\frac{a}{z^3}\right)dz$$ single-valued? The answer is $a=-2$. I am **not asking** for a full solution; I just hope someone can give me a hint, as I have no idea how this problem can be solved. p.s. I am aware of a theorem stating that $$f(z)=\int_C\frac{\phi(x)}{x-z}dx$$ is an analytic function for continuous $\phi$ and $z$ outside $C$. However, I don’t see how this theorem can be applied. Moreover, analyticity is too strong in comparison to single-valued-ness.

**Hint:** \begin{align}f\text{ is single-valued}&\iff\text{for every loop }\gamma,\ \displaystyle\int_\gamma e^z\left(\frac 1z+\frac a{z^3}\right)\,\mathrm dz=0\\\&\iff\operatorname{res}_{z=0}\left(e^z\left(\frac 1z+\frac a{z^3}\right)\right)=0.\end{align}