What is $e, \pi, \ln 2,...$ etc in p adic?

Question

What is $e, \pi, \ln 2,...$ etc in p adic? And how to flip digits of decimal points? Does p-adic have their own constants? 10 adic base.

Answer 1

The $p$-adic logarithm function and $p$-adic exponential function have been defined. The value of $\ln 2$ can be worked out, although it's a little tricky. The value of $e$ itself can't be worked out, I don't think - the series doesn't converge there - but $e^p$ can, for example.

I don't know if there is a $p$-adic analogue of $\pi$.