The element $\displaystyle \exp(\lambda x)$ is by definition the sum of the power series $\displaystyle \textstyle\sum_{n=0}^\infty\lambda^nx^n/n!$. Thus $\displaystyle \varphi(\lambda) = \textstyle\sum_{n=0}^\infty\lambda^nf(x^n)/n!$. This is a scalar-valued power series in $\displaystyle \lambda$ with infinite radius of convergence, so it can be differentiated term by term and is analytic throughout $\displaystyle \mathbb{C}$. If in addition it never takes the value zero, then its logarithm $\displaystyle \psi(\lambda)$ can be defined as an entire function.