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.

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 \mathbb{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 \mathbb{C}_p , the completion of the algebraic closure of \mathbb{Q}_p , by \exp_p(z)=\sum_{n=0}^\infty\frac{z^n}{n!}. However, unlike exp which converges on all of \mathbb{C} , \exp_p only converges on the disc |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 |z|_p 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\neq 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 \mathbb{C}_p satisfying |x|_p and so defines the p-adic logarithm function \log_p(z) for |z-1|_p satisfying the usual property \log_p(zw)=\log_p(z)+\log_p(w) . The function \log_p can be extended to all of \mathbb{C} (the set of nonzero elements of \mathbb{C}_p ) by imposing that it continues to satisfy this last property and setting \log_p(p)=0 . Specifically, every element w of \mathbb{C} can be written as w=p^r\cdot\zeta\cdot z with r a rational number, \zeta a root of unity, and |z-1|_p , in which case \log_p(w)=\log_p(z) . This function on \mathbb{C} 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 to all of \mathbb{C} for each choice of \log_p(p) in \mathbb{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 \mathbb{C}_p then \log_p(zw)=\log_p(z)+\log_p(w) . 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 \mathbb{C}_p of the form p^r\cdot\zeta where r is a rational number and \zeta is a root of unity. Note that there is no analogue in \mathbb{C}_p of Euler's identity, e^{2\pi i}=1 . This is a corollary of Strassmann's theorem. Another major difference to the situation in \mathbb{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 . ==Notes== ==References== ==External links== p-adic exponential and p-adic logarithm

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