nLab reciprocal function

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Context

Arithmetic

Algebra

Contents

Definition

In Heyting fields

Given a Heyting field FF, let us define the type of all terms in FF apart from 0:

F #0{aF|a#0}F_{#0} \coloneqq \{a \in F \vert a # 0\}

The reciprocal or reciprocal function is a partial function 1:F #0F\frac{1}{-}:F_{#0} \to F such that for all aF #0a \in F_{#0} we have a1a=1a \cdot \frac{1}{a} = 1 and 1aa=1\frac{1}{a} \cdot a = 1

In a discrete field, the reciprocal is a function f:FF+1f:F \to F + 1 defined as

f(x)1xf(x) \coloneqq \frac{1}{x}

for x0x \neq 0 and

f(x)f(x) \coloneqq \bot

for x=0x = 0, where 1\bot \in 1. This is because for a discrete field FF, the set F #0F 0F_{#0} \simeq F_{\neq 0} is a decidable subset of FF.

In dense sequentially Cauchy complete ordered integral domains

Let RR be a ordered integral domain, and for all elements aRa \in R and bRb \in R let (a,b)(a, b) be the open subinterval containing all elements greater than aa and less than bb. Then the sequences

f(p)(x) n=0 px nf(p)(x) \coloneqq \sum_{n=0}^{p} x^n
g(p)(x) n=0 p(1) nx ng(p)(x) \coloneqq \sum_{n=0}^{p} (-1)^n x^n

indexed by natural number pp \in \mathbb{N} are Cauchy sequences for all elements x(1,1)x \in (-1, 1), and if RR is sequentially Cauchy complete, it has a limit for elements x(1,1)x \in (-1, 1) as

f (x)lim p n=0 px nf_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} x^n
g (x)lim p n=0 p(1) nx ng_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} (-1)^n x^n

If RR is also dense with given element a(0,1)a \in (0, 1), then there are sequences of sequences

f(i)(p)(x)a i n=0 p(a i) n(xf (a i+1)) nf'(i)(p)(x) \coloneqq a^i \sum_{n=0}^{p} (-a^i)^n (x-f_\infty(-a^i+1))^n
g(i)(p)(x)a i n=0 p(a i) n(x+g (a i1)) ng'(i)(p)(x) \coloneqq a^i \sum_{n=0}^{p} (-a^i)^n (x+g_\infty(a^i-1))^n

indexed by natural numbers ii \in \mathbb{N} and pp \in \mathbb{N}, the first which is Cauchy for elements x(2f (a i+1),0)x \in (-2 f_\infty(-a^i+1), 0) and the second which is Cauchy for x(0,2g (a i1))x \in (0, 2 g_\infty(a^i-1)). Since RR is sequentially Cauchy complete, both have limits as

f (i)(x)lim pa i n=0 p(a i) n(xf (a i+1)) nf_\infty'(i)(x) \coloneqq \lim_{p \to \infty} a^i \sum_{n=0}^{p} (-a^i)^n (x-f_\infty(-a^i+1))^n
g (i)(x)lim pa i n=0 p(a i) n(x+g (a i1)) ng_\infty'(i)(x) \coloneqq \lim_{p \to \infty} a^i \sum_{n=0}^{p} (-a^i)^n (x+g_\infty(a^i-1))^n

which themselves are Cauchy, and thus have limits

f(x)lim if (i)(x)f' '(x) \coloneqq \lim_{i \to \infty} f_\infty'(i)(x)
g(x)lim ig (i)(x)g' '(x) \coloneqq \lim_{i \to \infty} g_\infty'(i)(x)

Since both f (a i+1)f_\infty(-a^i+1) and g (a i1)g_\infty(a^i-1) go to infinity as ii goes to infinity, the domain of ff' ' is (,0)(-\infty, 0) and the domain of gg' ' is (0,)(0, \infty).

The reciprocal 1():(,0)(0,)R\frac{1}{(-)}:(-\infty, 0)\union (0, \infty) \to R is a piecewise defined partial function defined as

1x{f(x) x(,0) g(x) x(0,) \frac{1}{x} \coloneqq \begin{cases} f' '(x) & x \in (-\infty, 0) \\ g' '(x) & x \in (0, \infty) \end{cases}

Thus, every dense sequentially Cauchy complete ordered integral domain is an ordered field.

In sequentially Cauchy complete ordered integral rational algebras

Let RR be a ordered integral domain which is a \mathbb{Q} -algebra, and for all elements aRa \in R and bRb \in R let (a,b)(a, b) be the open subinterval containing all elements greater than aa and less than bb. Then the sequence

g(p)(x) n=0 p(1) nx nn+1g(p)(x) \coloneqq \sum_{n=0}^{p} \frac{(-1)^n x^n}{n + 1}

indexed by natural number pp \in \mathbb{N} is a Cauchy sequence for all elements x(1,1)x \in (-1, 1), and if RR is sequentially Cauchy complete, it has a limit for elements x(1,1)x \in (-1, 1) as

g (x)lim p n=0 p(1) nx nn+1g_\infty(x) \coloneqq \lim_{p \to \infty} \sum_{n=0}^{p} \frac{(-1)^n x^n}{n + 1}

There is a sequence of sequences

g(i)(p)(x)i n=0 p(i) n(x+g (i1)) nn+1g'(i)(p)(x) \coloneqq i \sum_{n=0}^{p} \frac{(-i)^n (x+g_\infty(i-1))^n}{n + 1}

indexed by natural numbers ii \in \mathbb{N} and pp \in \mathbb{N}, which is Cauchy for x(0,2g (i1))x \in (0, 2 g_\infty(i-1)).

Since RR is sequentially Cauchy complete, the function has a limit as

g (i)(x)lim pi n=0 p(i) n(x+g (i1)) nn+1g_\infty'(i)(x) \coloneqq \lim_{p \to \infty} i \sum_{n=0}^{p} \frac{(-i)^n (x+g_\infty(i-1))^n}{n + 1}

which itself is Cauchy, and thus has a limit

ln(x)lim ig (i)(x)\ln(x) \coloneqq \lim_{i \to \infty} g_\infty'(i)(x)

called the natural logarithm. Since g (a i1)g_\infty(a^i-1) goes to infinity as ii goes to infinity, the domain of ln(x)\ln(x) is (0,)(0, \infty).

The exponential function is defined as as

exp(x)lim n i=0 nx ii!\exp(x) \coloneqq \lim_{n \to \infty} \sum_{i = 0}^{n} \frac{x^i}{i!}

and the reciprocal function is defined as

1x{exp(ln(x)) x(,0) exp(ln(x)) x(0,) \frac{1}{x} \coloneqq \begin{cases} -\exp(- \ln(-x)) & x \in (-\infty, 0) \\ \exp(- \ln(x)) & x \in (0, \infty) \end{cases}

See also

Last revised on August 21, 2024 at 01:54:41. See the history of this page for a list of all contributions to it.