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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, particularly `F33f`_`[p-adic analysis`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=P-adic_analysis]`_`f, the `!`*p`*-adic exponential function`! is a `*p`*-adic analogue of the usual `F33f`_`[exponential function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_function]`_`f on the `F33f`_`[complex numbers`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_numbers]`_`f. As in the complex case, it has an inverse function, named the `!`*p`*-adic logarithm`!.

>>Contents

• `F0af`_`[Definition`#definition]`_`f

• `F0af`_`[p -adic logarithm function`#p-adic-logarithm-function]`_`f

• `F0af`_`[Properties`#properties]`_`f

• `F0af`_`[Notes`#notes]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[Citations`#citations]`_`f

• `F0af`_`[List of references`#list-of-references]`_`f

• `F0af`_`[External links`#external-links]`_`f

-─

>>Definition

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

exp ⁡ ⁡ ( z ) = ∑ ∑ n = 0 ∞ ∞ z n n ! . {\\displaystyle \\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 ) = ∑ ∑ n = 0 ∞ ∞ z n n ! . {\\displaystyle \\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 ) . {\\displaystyle |z|_{p}<p^{-1/(p-1)}.}

This is because `*p`*-adic series converge `F33f`_`[if and only if`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=If_and_only_if]`_`f 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 `F33f`_`[Legendre's formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Legendre's_formula]`_`f that if | z | p < p − − 1 / ( p − − 1 ) {\\displaystyle |z|_{p}<p^{-1/(p-1)}} then z n n ! {\\displaystyle {\\frac {z^{n}}{n!}}} tends to 0 {\\displaystyle 0} , `*p`*-adically.

Although the `*p`*-adic exponential is sometimes denoted `*e`*`*x`*, the `F33f`_`[number e`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=E_(mathematical_constant)]`_`f 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,`:cite-ref-1[`F5bf`_`[a`#cite-note-1]`_`f] but there are multiple such roots and there is no canonical choice among them.`:cite-ref-2[`F5bf`_`[1`#cite-note-2]`_`f]

>>p -adic logarithm function

The power series

log p ⁡ ⁡ ( 1 + x ) = ∑ ∑ n = 1 ∞ ∞ ( − − 1 ) n + 1 x n n , {\\displaystyle \\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`*p`*`*z`* + log`*p`*`*w`*. 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`* = `*pr`*·ζ·`*z`* with `*r`* a `F33f`_`[rational number`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_number]`_`f, ζ a `F33f`_`[root of unity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Root_of_unity]`_`f, and |`*z`* − 1|`*p`* < 1,`:cite-ref-3[`F5bf`_`[2`#cite-note-3]`_`f] in which case log`*p`*(`*w`*) = log`*p`*(`*z`*).`:cite-ref-4[`F5bf`_`[b`#cite-note-4]`_`f] 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`*.`:cite-ref-5[`F5bf`_`[3`#cite-note-5]`_`f]

>>Properties

If `*z`* and `*w`* are both in the `F33f`_`[radius of convergence`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Radius_of_convergence]`_`f 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`*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 `!C`!`*p`* of the form `*pr`*·ζ where `*r`* is a rational number and ζ is a root of unity.`:cite-ref-6[`F5bf`_`[4`#cite-note-6]`_`f]

Note that there is no analogue in `!C`!`*p`* of `F33f`_`[Euler's identity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler's_identity]`_`f, `*e`*2`*πi`* = 1. This is a corollary of `F33f`_`[Strassmann's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Strassmann's_theorem]`_`f.

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 `F33f`_`[Artin–Hasse exponential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Artin–Hasse_exponential]`_`f — can be used instead which converges on |`*z`*|`*p`* < 1.

>>Notes

`:cite-note-1`!a.`! `F0af`_`[↑`#cite-ref-1]`_`f or a 4th root of exp2(4), for `*p`* = 2

`:cite-note-4`!b.`! `F0af`_`[↑`#cite-ref-4]`_`f 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.

>>References

>>>Citations

`:cite-note-2`!1.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Robert 2000`#citerefrobert2000]`_`f, p. 252

`:cite-note-3`!2.`! `F0af`_`[↑`#cite-ref-3]`_`f `F33f`_`[Cohen 2007`#citerefcohen2007]`_`f, Proposition 4.4.44

`:cite-note-5`!3.`! `F0af`_`[↑`#cite-ref-5]`_`f `F33f`_`[Cohen 2007`#citerefcohen2007]`_`f, §4.4.11

`:cite-note-6`!4.`! `F0af`_`[↑`#cite-ref-6]`_`f `F33f`_`[Cohen 2007`#citerefcohen2007]`_`f, Proposition 4.4.45

>>>List of references

• Chapter 12 of `:citerefcassels1986`a`F33f`_`[Cassels, J. W. S.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=J._W._S._Cassels]`_`f (1986). `*Local fields`*. `F33f`_`[London Mathematical Society Student Texts`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=London_Mathematical_Society]`_`f. `F33f`_`[Cambridge University Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cambridge_University_Press]`_`f. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-521-31525-5.

• `:citerefcohen2007`a`F33f`_`[Cohen, Henri`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Henri_Cohen_(number_theorist)]`_`f (2007), `*Number theory, Volume I: Tools and Diophantine equations`*, `F33f`_`[Graduate Texts in Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Graduate_Texts_in_Mathematics]`_`f, vol. 239, New York: Springer, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/978-0-387-49923-9, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-387-49922-2, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2312337

• `:citerefrobert2000`aRobert, Alain M. (2000), `*A Course in `*p`*-adic Analysis`*, Springer, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-387-98669-3

>>External links

• p-adic exponential and p-adic logarithm

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