# 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.

## Definition

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

${\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}$

Entirely analogously, one defines the exponential function on Cp, the completion of the algebraic closure of Qp, by

${\displaystyle \exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}$

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

${\displaystyle |z|_{p}

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} then ${\displaystyle {\frac {z^{n}}{n!}}}$ tends to ${\displaystyle 0}$, 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,[a] but there are multiple such roots and there is no canonical choice among them.[1]

## p-adic logarithm function

The power series

${\displaystyle \log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},}$

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,[2] in which case logp(w) = logp(z).[b] 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.[3]

## 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.[4]

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

1. ^ or a 4th root of exp2(4), for p = 2
2. ^ 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.

## References

1. ^ Robert 2000, p. 252
2. ^ Cohen 2007, Proposition 4.4.44
3. ^ Cohen 2007, §4.4.11
4. ^ Cohen 2007, Proposition 4.4.45
• 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