I have a complex-valued random variable $X$ and I know the formula for $E\left[e^{\phi X}\right]$, for any $\phi\in\mathbb{C}$. It also happens that $P[Im(X)=y]=P[Im(X)=-y]$, i.e the probability density of the imaginary part of $X$ is symmetric around zero.
Based on this information, is there a way to calculate $E\left[e^{\phi Re(X)}\right]$?