Definition:Power (Algebra)/Real Number

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Definition

Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $r \in \R$ be a real number.


Definition 1

We define $x^r$ as:

$x^r := \map \exp {r \ln x}$

where $\exp$ denotes the exponential function.


Definition 2

Let $f : \Q \to \R$ be the real-valued function defined as:

$f \left({ q }\right) = x^q$

where $a^q$ denotes $a$ to the power of $q$.


Then we define $x^r$ as the unique continuous extension of $f$ to $\R$.


Definition 3

First let $x > 1$.

Let $r$ be expressed by its decimal expansion:

$r = n \cdotp d_1 d_2 d_3 \ldots$


For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as:

\(\displaystyle \psi_1\) \(=\) \(\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k}\) $\quad$ $\quad$
\(\displaystyle \psi_2\) \(=\) \(\displaystyle \psi_1 + \dfrac 1 {10^k}\) $\quad$ $\quad$


Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as:

$\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$.


Next let $x < 1$.

Then $x^r$ is defined as:

$x^r := \left({\dfrac 1 x}\right)^{-r}$


Finally, when $x = 1$:

$x^r = 1$


Complex Number

This definition can be extended to complex $r$:


Let $x \in \R$ be a real number such that $x > 0$.

Let $r \in \C$ be any complex number.

Then we define $x^r$ as:

$x^r := \map \exp {r \ln x}$

where $\exp$ denotes the complex exponential function.


Examples

Euler's Number to Power of Itself

$e^e \approx 15 \cdotp 15426 \, 22414 \, 79264 \, 18976 \, 0430 \ldots$

This sequence is A073226 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Euler's Number to Power of its Negative

$e^{-e} \approx 0 \cdotp 06598 \, 80358 \, 45312 \ldots$

This sequence is A073230 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Euler's Number to Power of its Reciprocal

$e^{1/e} \approx 1 \cdotp 44466 \, 78610 \, 09766 \, 13365 \, 83 \ldots$

This sequence is A073229 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Euler's Number to Power of Euler-Mascheroni Constant

$e^\gamma \approx 1 \cdotp 78107 \, 24179 \, 90197 \, 9852 \ldots$

This sequence is A073004 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Also see

  • Results about powers can be found here.