How could one port elementary functions to fields of characteristic $p$? Is the concept of a power series similar?

Question

What is the proper way to port familiar (say, real) functions such as $e^x$, $\sin(x)$, $\log(x)$, $\sqrt{x}$ and such to a field $F$ of characteristic $p>0$? For example, we could define $\exp:\mathbb{R}\to\mathbb{R},$ $x\mapsto \sum_{n\ge 0}\frac{x^n}{n!}$; is there a corresponding object $\exp_p:F \to F$, and how is it defined?

I am moderately familiar with finite fields of prime characteristic and am aware of but certainly less familiar with infinite fields of prime characteristic. Naively, since many elementary functions are defined by their Maclaurin series, perhaps one defines a formal power series in the same way; however, I'm concerned about division by $p$ or that the finite characteristic forces all such power series to reduce to polynomials. Apologies if this is unclear, I can try to clarify my question if needed.

Answer 1

I will mainly talk about $\exp$. For the others: $\sin$ can be obtained from $\exp$, $\log$ is the inverse of $\exp$, and $\sqrt x$ (or rather $\sqrt{1 - x}$) exists when characteristic is not $2$.

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The power series $\exp(x)$ is useful mainly because it satisfies a differential equation $\exp' = \exp$. It is (up to scalar) the unique power series which satisfies this equation.

Now in characteristic $p$, it is easy to see that there is no (nonzero) power series $f$ satisfying $f' = f$ (exercise).

This should suggest that there is nothing "as good as in characteristic $0$".

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Nevertheless, there are some "slightly worse" replacements, which lose one property or another.

I will mention just the following variant. Instead of using formal power series, which are the "ordinary generating functions" of the coefficient sequence, we may use "exponential generating functions".

In a general setting, when we have a commutative ring $R$, we may define the ring of "formal exponential series" $R\{ x\}$ (my notation, probably nonstandard).

The elements of $R\{ x\}$ are infinite sequences $(a_0, a_1, \dots)$ of elements of $R$. Addition and multiplication are defined as follows:

• $(a_n)_n + (b_n)_n = (a_n + b_n)_n$;
• $(a_n)_n \cdot (b_n)_n = (\sum_{i + j = n}\binom n i a_ib_j)_n$.

If every nonzero integer is invertible in $R$ (which is the case when $R$ is a field of characteristic $0$), then the ring $R\{ x\}$ is isomorphic to the ring of formal power series $R[[x]]$ by identifying $(a_n)_n \in R\{x\}$ with $\sum_n \frac{a_n}{n!} x^n\in R[[x]]$.

Thus the sequence $(1, 1, \dots) \in R\{x\}$ is, in characteristic $0$, identified with $\exp(x)$.

Now the same sequence exists in any characteristic, and can be viewed as a replacement of $\exp$ in certain applications.