P-adic exponential function explained

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

	infty
\exp(z)=\sum
	n=0

	zⁿ
	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

infty	zⁿ
	n!

n=0

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

	-1/(p-1)
\|z\|
	p<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

|z|_p<p^-1/(p-1)

then

	zⁿ
	n!

tends to

, 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 . It is possible to choose a number e to be a p-th root of exp_p(p) for, 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

	infty

	n=1

	(-1)ⁿ⁺¹xⁿ
	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 (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 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 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 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.

References

Chapter 12 of Book: Cassels, J. W. S. . J. W. S. Cassels

. J. W. S. Cassels . Local fields . . . 1986 . 0-521-31525-5 .

External links

p-adic exponential and p-adic logarithm