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p-adic exponential function

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In <a href="Mathematics" title="Mathematics">mathematics</a>, particularly <a href="P-adic_analysis" title="P-adic analysis">p-adic analysis</a>, the p-adic exponential function is a p-adic analogue of the usual <a href="Exponential_function" title="Exponential function">exponential function</a> on the <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a>. As in the complex case, it has an inverse function, named the p-adic logarithm.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}</annotation>

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</math><img src="./cdf70a282ef5a8ac58571961993736e837ce0ebe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.082ex; height:6.843ex;" alt="{\displaystyle \exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}" loading="lazy"></dd></dl>

Entirely analogously, one defines the exponential function on Cp, the completion of the algebraic closure of Qp, by

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<annotation encoding="application/x-tex">{\displaystyle \exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}</annotation>

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</math><img src="./bfdffb62154769b6e2fad406dd49eb81694e898a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.141ex; height:6.843ex;" alt="{\displaystyle \exp _{p}(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}.}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle |z|_{p}<p^{-1/(p-1)}.}</annotation>

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</math><img src="./bb6856fd689398c1b9b296366e47a70fa61aa602.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.717ex; height:3.676ex;" alt="{\displaystyle |z|_{p}<p^{-1/(p-1)}.}" loading="lazy"></dd></dl>

This is because p-adic series converge <a href="If_and_only_if" title="If and only if">if and only if</a> 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 <a href="Legendre's_formula" title="Legendre's formula">Legendre's formula</a> that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|_{p}<p^{-1/(p-1)}}">

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<annotation encoding="application/x-tex">{\displaystyle |z|_{p}<p^{-1/(p-1)}}</annotation>

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</math><img src="./ecbf993e62285d17aab3b9e4e1b9cc7fc4304d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.07ex; height:3.676ex;" alt="{\displaystyle |z|_{p}<p^{-1/(p-1)}}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {z^{n}}{n!}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {z^{n}}{n!}}}</annotation>

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</math><img src="./56fdf198cc22145adbb53f787b0f5e3e6f57d1ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.145ex; height:5.343ex;" alt="{\displaystyle {\frac {z^{n}}{n!}}}" loading="lazy"> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">, p-adically.

Although the p-adic exponential is sometimes denoted ex, the <a href="E_(mathematical_constant)" title="E (mathematical constant)">number e</a> itself has no p-adic analogue. This is because the power series expp(x) does not converge at x = 1. It is possible to choose a number e to be a p-th root of expp(p) for p ≠ 2,<a href="#cite_note-1">[a]</a> but there are multiple such roots and there is no canonical choice among them.<a href="#cite_note-2">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="p-adic_logarithm_function">p-adic logarithm function</h2></div>

The power series

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<annotation encoding="application/x-tex">{\displaystyle \log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},}</annotation>

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</math><img src="./fca2e19e1ac60715f23c1a31a74fdb436d406af5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.143ex; height:7.009ex;" alt="{\displaystyle \log _{p}(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}x^{n}}{n}},}" loading="lazy"></dd></dl>

converges for x in Cp satisfying |x|p < 1 and so defines the p-adic logarithm function logp(z) for |z − 1|p < 1 satisfying the usual property logp(zw) = logpz + logpw. The function logp can be extended to all of C× p  (the set of nonzero elements of Cp) by imposing that it continues to satisfy this last property and setting logp(p) = 0. Specifically, every element w of C× p  can be written as w = pr·ζ·z with r a <a href="Rational_number" title="Rational number">rational number</a>, ζ a <a href="Root_of_unity" title="Root of unity">root of unity</a>, and |z − 1|p < 1,<a href="#cite_note-3">[2]</a> in which case logp(w) = logp(z).<a href="#cite_note-4">[b]</a> This function on C× p  is sometimes called the Iwasawa logarithm to emphasize the choice of logp(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 logp(p) in Cp.<a href="#cite_note-5">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

If z and w are both in the <a href="Radius_of_convergence" title="Radius of convergence">radius of convergence</a> for expp, then their sum is too and we have the usual addition formula: expp(z + w) = expp(z)expp(w).

Similarly if z and w are nonzero elements of Cp then logp(zw) = logpz + logpw.

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

The roots of the Iwasawa logarithm logp(z) are exactly the elements of Cp of the form pr·ζ where r is a rational number and ζ is a root of unity.<a href="#cite_note-6">[4]</a>

Note that there is no analogue in Cp of <a href="Euler's_identity" title="Euler's identity">Euler's identity</a>, e2πi = 1. This is a corollary of <a href="Strassmann's_theorem" title="Strassmann's theorem">Strassmann's theorem</a>.

Another major difference to the situation in C is that the domain of convergence of expp is much smaller than that of logp. A modified exponential function — the <a href="Artin%E2%80%93Hasse_exponential" title="Artin–Hasse exponential">Artin–Hasse exponential</a> — can be used instead which converges on |z|p < 1.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> or a 4th root of exp2(4), for p = 2

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<li id="cite_note-4"><a href="#cite_ref-4">^</a> In factoring w as above, there is a choice of a root involved in writing pr since r is rational; however, different choices differ only by multiplication by a root of unity, which gets absorbed into the factor ζ.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3></div>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFRobert2000">Robert 2000</a>, p. 252

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFCohen2007">Cohen 2007</a>, Proposition 4.4.44

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFCohen2007">Cohen 2007</a>, §4.4.11

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFCohen2007">Cohen 2007</a>, Proposition 4.4.45

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<div class="mw-heading mw-heading3"><h3 id="List_of_references">List of references</h3></div>

<ul><li>Chapter 12 of <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><cite id="CITEREFCassels1986" class="citation book cs1"><a href="J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J. W. S.</a> (1986). Local fields. <a href="London_Mathematical_Society" title="London Mathematical Society">London Mathematical Society Student Texts</a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-31525-5</bdi>.</cite></li>

<li><cite id="CITEREFCohen2007" class="citation cs2"><a href="Henri_Cohen_(number_theorist)" title="Henri Cohen (number theorist)">Cohen, Henri</a> (2007), Number theory, Volume I: Tools and Diophantine equations, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 239, New York: Springer, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-387-49923-9">10.1007/978-0-387-49923-9</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-49922-2</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2312337">2312337</a></cite></li>

<li><cite id="CITEREFRobert2000" class="citation cs2">Robert, Alain M. (2000), A Course in p-adic Analysis, Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-98669-3</bdi></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/PadicExponentialAndPadicLogarithm">p-adic exponential and p-adic logarithm</a></li></ul></div><div><div class="zim-footer">

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