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.

Definition

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

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

Entirely analogously, one defines the exponential function on C_p, the completion of the algebraic closure of Q_p, by

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

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

$|z|_{p}<p^{-1/(p-1)}.$

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

p-adic logarithm function

One can also define a p-adic logarithm function by the power series

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

for |z − 1|_p < 1. But this can be extended to all nonzero elements of C_p by writing any z in C_p as z = p^m·u·v, where m is a rational number, u is a root of unity of order coprime to p, and v lies in the original domain of convergence for log_p, so |v − 1|_p < 1. We then define log_p(z) to be log_p(v).

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.

And for suitable z, so that everything is defined, we have exp_p(log_p(z)) = z and log_p(exp_p(z)) = z.

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.

References

• Cassels, J. W. S. (1986). Local fields. London Mathematical Society Student Texts. Cambridge University Press. ISBN 0-521-31525-5.

External links

• "p-adic exponential and p-adic logarithm". PlanetMath.