The wikipedia page for Elementary Function defines it as follows.

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).

It goes on to say the following about trigonometric functions.

The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms.

However, the trigonometric function definitions I have seen do not appear to satisfy the 'Elementary Function' requirements. Here are two definitions I've seen alongside my objections on why the definition does not appear to be Elementary.

The relevant section of the Trigonometric Functions wikipedia page includes the following definitions of $\sin$ and $\cos$.

$$\cos x = \operatorname{Re}(e^{i x})$$ $$\sin x = \operatorname{Im}(e^{i x})$$

  • My objection: the definition uses the $\operatorname{Re}$ and $\operatorname{Im}$ operators, which do not appear to be Elementary Functions themselves according to the definition.

Also, the Wolfram Research Functions site lists the following sum as a definition for $\sin$.

$$\sin z = \sum_{k=0}^\infty \frac{(-1)^k z^{2 k + 1}}{(2 k + 1)!}$$

  • My objection: an infinite number of operations are used, which is in violation of the definition.

Is the wikipedia article correct in asserting that the trigonometric functions are indeed Elementary?

If so, how can one construct the trigonometric functions using a finite number of compositions of elementary operations?

A side note:

In addition to the objections listed above, it seems strange that $e$ -- being transcendental and therefore not constructible using finite compositions of the elementary operations -- could be used as a constant in the construction of other elementary operations.

I would be interested in hearing perspective on whether or why transcendental constants are permitted in the construction of Elementary Functions.

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    $\begingroup$ Possible duplicate of What makes elementary functions elementary? $\endgroup$ – Mr. T Oct 26 '17 at 20:51
  • $\begingroup$ To summarize Mr. T's comment as it relates to your question, the functions $e^x$ and $\log(x)$ are also considered elementary, and all trig functions can be written in terms of these functions. $\endgroup$ – Carl Schildkraut Oct 26 '17 at 20:57
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    $\begingroup$ It might make things clearer to write $\cos z=\frac{e^{iz}+e^{-iz}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$. $\endgroup$ – carmichael561 Oct 26 '17 at 21:00
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    $\begingroup$ The first sentence of a Wikipedia article doesn't necessarily provide a technically precise definition, but rather "should informally define or describe the subject." en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/… $\endgroup$ – PersonX Oct 27 '17 at 13:12

Let $\mathbb{F}$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. Let $\mathcal{P}(\mathbb{F})$ the set of $\mathbb{F}$-valued power functions ()functions of the form $x \mapsto x^{\alpha}$), $\mathcal{EL(\mathbb{F})}$ the set of $\mathbb{F}$-valued exponential and logarithmic functions and $\mathcal{T}(\mathbb{F})$ the set of $\mathbb{F}$-valued trigonometric functions. I think the class of elementary functions as the minimal $\mathcal{E}=(\mathcal{E},+,\cdot, \circ)$ set such that $\mathcal{P}(\mathbb{F}),\mathcal{EL(\mathbb{F})},\mathcal{T}(\mathbb{F}) \subseteq \mathcal{E}$ and $\mathcal{E}$ is closed unde addition, substraction, multiplication, division and compsoition of functions (so the set if polynomials $\mathbb{F}[x] \in \mathcal{E}$ and the set of hiperbolic functions $\mathcal{H}(\mathbb{F})$ also lives in $\mathcal{E}$).

If you think of the complex exponential function $\exp:z \to e^z$ as an elementary function then $\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2}$ and $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$, so $\sin, \cos \in \mathcal{E}$

  • $\begingroup$ You should use z everywhere in the definition of $\sin z$ and $\cos z$. $\endgroup$ – Bernard Oct 26 '17 at 21:21
  • $\begingroup$ @ Bernard wow thanks! Typo ;P $\endgroup$ – James Garrett Oct 26 '17 at 21:32
  • $\begingroup$ I love typos too ;o) $\endgroup$ – Bernard Oct 26 '17 at 21:35

Liouville defined the set of Elementary functions with means of algebra. Ritt wrote in [Ritt 1925]: "The elementary functions are understood here to be those which are obtained in a finite number of steps by performing algebraic operations and taking exponentials and logarithms."

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

Considering $\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2}$ and $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$, the trigonometric functions belong quite obviously to the set of Elementary functions.

The constants in the set "Elementary functions" are the real or complex numbers.


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