In this spherical harmonics paper, there's a periodic function $\Phi(\phi)$ defined as
$$\Phi(\phi)= \bigg\{ \begin{array} \\e^{im\phi} \\e^{-im\phi} \end{array} \space m = 0,1,2,3...$$
Is $e^{im\phi}$ the same thing as the more conventional $e^{ix}$?
Answer: $e^{i m \phi}$ is the same thing as $e^{i x}$ when $x = m \phi$. Note that $i m \phi$ stands for the product $i \, m \, \phi$ and not the imaginary part of $\phi$ here.
This should be interpreted as a family of periodic functions $e^{i m \phi}$ and $e^{-i m \phi}$ indexed by all non-negative integers $m = 0, 1, 2, \dots$.