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.
Definition
The usual exponential function on C is defined by the infinite series
exp ( z ) = ∑ n = 0 ∞ z n n ! . {\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}Entirely analogously, one defines the exponential function on Cp, the completion of the algebraic closure of Qp, by
exp p ( z ) = ∑ n = 0 ∞ z n n ! . {\displaystyle \exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}However, unlike exp which converges on all of C, expp only converges on the disc
| z | p < p − 1 / ( p − 1 ) . {\displaystyle |z|_{p}<p^{-1/(p-1)}.}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 | z | p < p − 1 / ( p − 1 ) {\displaystyle |z|_{p}<p^{-1/(p-1)}} then z n n ! {\displaystyle {\frac {z^{n}}{n!}}} tends to 0 {\displaystyle 0} , p-adically.
Although the p-adic exponential is sometimes denoted ex, the number e itself has no p-adic analogue. This is because the power series expp(x) does not converge at x = 1. It is possible to choose a number e to be a p-th root of expp(p) for p ≠ 2,1 but there are multiple such roots and there is no canonical choice among them.2
p-adic logarithm function
The power series
log p ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 x n n , {\displaystyle \log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},}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 continues 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 < 1,3 in which case logp(w) = logp(z).4 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.5
Properties
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.
For z in the domain of expp, we have expp(logp(1+z)) = 1+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.6
Note that there is no analogue in Cp of Euler's identity, e2πi = 1. This is a corollary of Strassmann's theorem.
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.
Notes
- 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, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337
- Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, ISBN 0-387-98669-3
External links
References
or a 4th root of exp2(4), for p = 2 ↩
Robert 2000, p. 252 - Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, ISBN 0-387-98669-3 ↩
Cohen 2007, Proposition 4.4.44 - Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337 https://doi.org/10.1007%2F978-0-387-49923-9 ↩
In factoring w as above, there is a choice of a root involved in writing pr since r is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ. ↩
Cohen 2007, §4.4.11 - Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337 https://doi.org/10.1007%2F978-0-387-49923-9 ↩
Cohen 2007, Proposition 4.4.45 - Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 239, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337 https://doi.org/10.1007%2F978-0-387-49923-9 ↩