Elementary and Closed Form Functions
Idea
Informally, a function being
elementary implies that it's a relatively simple arithmetical operation. Similarly, a function having a
closed-form implies its equivalence to some (often simpler) combination of elementary operations.
Definition
In the context of this site
[1][1] Being dependent on context, there is no universally accepted definition for the terms elementary or closed-form., we define the
elementary functions according to a minimal construction. Over
$\mathbb{C}$,
- The identity function is considered elementary,
- Any constant function is considered elementary,
- $f+g$ is considered elementary,
- $-f$ is considered elementary,
- $\exp f$ is considered elementary,
- $\ln f$ is considered elementary (under any choice of branch),
for all elementary functions $f$ and $g$. Finally, we consider an expression to be in
closed-form if it can be equivalently expressed as a finite combination of elementary functions, even if the number of operations grows (perhaps unbounded) with the variation of its parameters.
Corollaries
We may realize that our definition implies the following operations are also elementary.
- Multiplication: $fg=\exp(\ln f + \ln g)$.
- Exponentiation: $f^g=\exp(g\ln f)$.
- Reciprocals: $\frac{1}{g}=\exp(-\ln f)$.
- Division: $\frac{f}{g} = f\frac{1}{g}$.
- Roots: $\sqrt[g]{f}=f^{\frac{1}{g}}$.
- Logarithms: $\log_g f = \frac{\ln f}{\ln g}$.
- Trigonometric functions: e.g; $\cos z = \frac{\exp (i z)+\exp (-i z)}{2}$.
There are two important differences in our definitions of elementary and closed-form.
- Something can have a closed-form but not be elementary. (E.g; the taylor series of $\cos$ is not elementary, but it has a closed-form in terms of complex exponentiation). Basically, $\text{elementary}\implies\text{closed-form}$ but not conversely, necessarily.
- Elementary functions have a fixed number of operations, whereas the number of operations in a closed-form function may vary with respect to its parameters (e.g; tetration has a closed-form as a finite number of exponentations, but the number of exponentiations varies with respect to its height).
Ability of branch choice
The idea that any choice of branch of the logarithm is included in the definition is perhaps controversial, for it allows for some powerful functions to be considered elementary. For instance,
- If we choose our branches such that $0\le \frac{1}{i}\ln e^{i\theta} < 2\pi$, then we may define the modulo operator $x\!\mod y:= \frac{y}{2\pi i}\ln e^{\frac{2\pi i x}{y}}$, which is elementary, by our definition.
- We may then define the floor function as $[x]:=x-(x\!\mod 1)$.
- Similarly, the absolute value function may be defined as $x=\sqrt{x^2}$ for real numbers.
Interestingly, this ability (tied to the properties of complex numbers) inherently gives rise to the construction of
arithmetic indicators.
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