P-adic exponential function

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

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

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

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

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 large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if then $\backslash frac\{z^n\}\{n!\}$ tends to $0$, p-adically.

Although the p-adic exponential is sometimes denoted e^x, the number e itself has no p-adic analogue. This is because the power series exp_p(x) does not converge at {{nowrap|x {{=}} 1}}. It is possible to choose a number e to be a p-th root of exp_p(p) for {{nowrap|p ≠ 2}},{{efn|or a 4th root of exp₂(4), for {{nowrap|p {{=}} 2}}}} but there are multiple such roots and there is no canonical choice among them.{{harvnb|Robert|2000|p=252}}

The power series

:$\backslash log\_p(1+x)=\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}x^n\}\{n\},$

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 {{SubSup|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 {{SubSup|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,{{harvnb|Cohen|2007|loc=Proposition 4.4.44}} in which case log_p(w) = log_p(z).{{efn|In factoring w as above, there is a choice of a root involved in writing p^r since r is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.}} This function on {{SubSup|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 {{SubSup|C|p|×}} for each choice of log_p(p) in C_p.{{harvnb|Cohen|2007|loc=§4.4.11}}

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.

For z in the domain of exp_p, we have exp_p(log_p(1+z)) = 1+z and log_p(exp_p(z)) = z.

The roots of the Iwasawa logarithm log_p(z) are exactly the elements of C_p of the form p^r·ζ where r is a rational number and ζ is a root of unity.{{harvnb|Cohen|2007|loc=Proposition 4.4.45}}

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.

{{Notelist}}

{{reflist}}

• Chapter 12 of {{cite book | last=Cassels | first=J. W. S. | authorlink=J. W. S. Cassels | title=Local fields | series=London Mathematical Society Student Texts | publisher=Cambridge University Press | year=1986 | isbn=0-521-31525-5 }}
• {{Citation

| last=Cohen

| first=Henri

| author-link=Henri Cohen (number theorist)

| title=Number theory, Volume I: Tools and Diophantine equations

| publisher=Springer

| location=New York

| series=Graduate Texts in Mathematics

| volume=239

| year=2007

| isbn=978-0-387-49922-2

| mr=2312337

| doi=10.1007/978-0-387-49923-9

}}

• {{Citation |last=Robert |first=Alain M. |year=2000 |title=A Course in p-adic Analysis |publisher=Springer |isbn=0-387-98669-3}}

• [https://planetmath.org/PadicExponentialAndPadicLogarithm p-adic exponential and p-adic logarithm]

Category:Exponentials

Category:p-adic numbers