P-adic exponential function
# 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

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

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 large p-adically, a small value of z is needed in the numerator. It follows from Legendre's formula that if $|z|_{p}<p^{-1/(p-1)}$ then ${\frac {z^{n}}{n!}}$ 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 x = 1. It is possible to choose a number e to be a p-th root of $exp_p(p)$ for p ≠ 2, but there are multiple such roots and there is no canonical choice among them.

## p-adic logarithm function

The power series

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

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 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 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, in which case $log_p(w)$ = $log_p(z).$ This function on 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 C × p for each choice of $log_p(p)$ in $C_p.$

## 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 $log_p(zw)$ = $log_pz$ + $log_pw.$

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.

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.

