P-adic exponential function

In mathematics, particularly p-adic analysis, the p-adic exponential function is a p-adic analogue of the usual exponential function on the complex numbers. As in the complex case, it has an inverse function, named the p-adic logarithm.

The usual exponential function on C is defined by the infinite series

Entirely analogously, one defines the exponential function on C_p, the completion of the algebraic closure of Q_p, by

However, unlike exp which converges on all of C, exp_p only converges on the disc

This is because p-adic series converge if and only if the summands tend to zero, and since the n! in the denominator of each summand tends to make them large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if ${\displaystyle |z|_{p}<p^{-1/(p-1)}}$ then ${\displaystyle {\frac {z^{n}}{n!}}}$ tends to ${\displaystyle 0}$, p-adically.

Although the p-adic exponential is sometimes denoted e^x, the number e itself has no p-adic analogue. This is because the power series exp_p(x) does not converge at x = 1. It is possible to choose a number e to be a p-th root of exp_p(p) for p ≠ 2, but there are multiple such roots and there is no canonical choice among them.

The power series

converges for x in C_p satisfying |x|_p < 1 and so defines the p-adic logarithm function log_p(z) for |z − 1|_p < 1 satisfying the usual property log_p(zw) = log_pz + log_pw. The function log_p can be extended to all of C×
p  (the set of nonzero elements of C_p) by imposing that it continues to satisfy this last property and setting log_p(p) = 0. Specifically, every element w of C×
p  can be written as w = p^r·ζ·z with r a rational number, ζ a root of unity, and |z − 1|_p < 1, in which case log_p(w) = log_p(z). This function on C×
p  is sometimes called the Iwasawa logarithm to emphasize the choice of log_p(p) = 0. In fact, there is an extension of the logarithm from |z − 1|_p < 1 to all of C×
p  for each choice of log_p(p) in C_p.