P-adic logarithm

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

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.

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