I'm looking for ways to compute some integrals of continuous (or integrable) functions $f:\mathbb{Z}_p\to\mathbb{C}$, where by $\mathbb{Z}_p$ I denote the $p$-adic numbers for a fixed prime $p$.
Specifically, we know that $\mathbb{Z}_p$ can be made into a compact probability space using the Haar measure $m$ (that is unique if we impose the normalization condition $m(\mathbb{Z}_p) = 1$). Then, it makes sense to talk about $\int_{\mathbb{Z}_p}f(x)\,dm$. For example, in these notes , the integral of $f(x) = (\lvert x^d\rvert_p)^s$ is computed for fixed real numbers $s,d\geq0$ ($\lvert\cdot\rvert_p$ is the $p$-adic absolute value).
Other sources, I have found, prefer to deal with integrals of functions $\mathbb{Z}_p\to\mathbb{C}_p$ (like the Volkenborn integration theory). I hope to ask: are there more examples of integrals of maps $\mathbb{Z}_p\to\mathbb{C}$? Or, is there a way to turn (some, perhaps, but not all) continuous maps to $\mathbb{C}_p$ into complex-valued ones and define their integral that way? Thank you very much in advance.