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 Cp, the completion of the algebraic closure of Qp, by
However, unlike exp which converges on all of C, expp 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 Cp satisfying |x|p < 1 and so defines the p-adic logarithm function logp(z) for |z − 1|p < 1 satisfying the usual property logp(zw) = logpz + logpw. The function logp can be extended to all of C ×
p (the set of nonzero elements of Cp) by imposing that it continue to satisfy this last property and setting logp(p) = 0. Specifically, every element w of C ×
p can be written as w = pr·ζ·z with r a rational number, ζ a root of unity, and |z − 1|p < 1, in which case logp(w) = logp(z). This function on C ×
p is sometimes called the Iwasawa logarithm to emphasize the choice of logp(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 logp(p) in Cp.
If z and w are both in the radius of convergence for expp, then their sum is too and we have the usual addition formula: expp(z + w) = expp(z)expp(w).
Similarly if z and w are nonzero elements of Cp then logp(zw) = logpz + logpw.
And for suitable z, so that everything is defined, we have expp(logp(z)) = z and logp(expp(z)) = z.
The roots of the Iwasawa logarithm logp(z) are exactly the elements of Cp of the form pr·ζ where r is a rational number and ζ is a root of unity.
Another major difference to the situation in C is that the domain of convergence of expp is much smaller than that of logp. A modified exponential function — the Artin–Hasse exponential — can be used instead which converges on |z|p < 1.
- Chapter 12 of Cassels, J. W. S. (1986). Local fields. London Mathematical Society Student Texts. Cambridge University Press. ISBN 0-521-31525-5.
- Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337