P adic exponential function

Definition

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 very large p-adically, rather a small value of z is needed in the numerator.

p-adic logarithm function

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.

Properties

If z and w are both in the radius of convergence for exp_p, then their sum is too and we have the usual addition formula: exp_p(z + w) = exp_p(z)exp_p(w).

Similarly if z and w are nonzero elements of C_p then log_p(zw) = log_pz + log_pw.

For z in the domain of exp_p, we have exp_p(log_p(1+z)) = 1+z and log_p(exp_p(z)) = z.

The roots of the Iwasawa logarithm log_p(z) are exactly the elements of C_p of the form p^r·ζ where r is a rational number and ζ is a root of unity.

Note that there is no analogue in C_p of Euler's identity, e^2πi = 1. This is a corollary of Strassmann's theorem.

Another major difference to the situation in C is that the domain of convergence of exp_p is much smaller than that of log_p. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on |z|_p < 1.