What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, logarithmic, exponential, and $n$th roots, and solving problems that have solutions which are elementary functions. I've been curious why these functions are called elementary, as opposed to some other functions that turn up rather naturally in mathematics. What is the reason that these functions take up most of our attention, and is there a reason that some additional functions are not included amongst the elementary functions? In other words, what property or properties do these functions possess that separates them from non-elementary functions (if there is one)?

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Those elementary functions are the familiar functions since from lower level maths. They are the most useful functions regarding mathematics and they play a very vital role in applications. That's makes them elementary. –  Hassan Muhammad Mar 9 '12 at 5:22
Here is a reverse kind of question/discussion What is a special function –  AD. Mar 9 '12 at 5:30
The polynomial, exponential, sine, and cosine functions are "elementary" because they are very useful and will more frequently arise naturally in an investigation (whether within math or in an application) than most other functions. So everyone needs to know them. But why are they so useful? I think fundamentally it's because they are solutions to some of the simplest differential equations you could write down. The polynomials are the functions whose nth derivative is constantly 0. The sine and cosine functions satisfy $y'' + y = 0$ and the exponential function satisfies $y' = y$. –  Mike Benfield Apr 26 '12 at 13:06
@Mike: or if one wishes to be a bit more inclusive, the exponential function, the sine, the cosine, and their hyperbolic counterparts all satisfy the differential equation $y^{(iv)}=y$. –  Guess who it is. Apr 26 '12 at 13:31
As far as I can see, there's no reason to have a fixed, unchanging notion of "elementary function." More useful are statements like: "The least set of entities that includes the entities [...] and is closed under the operations [...], also includes the entities [...] and is also closed under the operations [...]. Furthermore, all entities in this class can be uniquely expressed in the form [...]." Substitute the word "entities" for the word "functions" and you'll get lots of interesting notions of "class of elementary functions." –  goblin Nov 4 '13 at 3:40

As Sivaram Ambikasaran mentioned the description on Wikipedia is fine.

I believe the class of elementary functions, $E$, is commonly thought of as a construction of the form

1. All polynomials are in $E$

2. The exponential and the logarithm function is in $E$

3. The sine and cosine functions are in $E$.

4. $E$ is closed under addition, subtraction, multiplication, division and composition (finitely many operations of these).

5. $E$ is the smallest set with the properties 1-4.

This applies to both real or complex valued functions.

Edit 1:

Some examples of functions that are elementary

• $f(x)=1-x^2$
• $s(x) = \sqrt{x}$ (see addendum below)
• $g(x)=\arctan x$ (see addendum below)
• $U(x)=\sin\frac{1}{\log(1+x^2)}$
• $A(x)=|x| = \sqrt{x^2}$
• $_2F_1(1,1,2,x) = \log(1-x)$ (a Gauss Hypergeometric notation that ends up in a elementary function)

Some examples of functions that are not elementary

• The Sine integral $\operatorname{si}(x)=\int_0^x\frac{\sin t}{t}dt$
• The Error function $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}dt$
• The Cantor function
• The characteristic function of an interval.

Edit 2:

The nomenclature elementary function is of course used since the functions that are elementary can be deduced using finitely many applications of elementary operations on basic "high school functions".

Edit 3:

Convinced by comments - removed closed under inversion in step 5. Added log in step 2.

To see that the square root function $s(x)=\sqrt{x}$ is elementary using the above definition just note that $$x\mapsto \frac{1}{2}\log x = \log x^{1/2}=\log\sqrt{x}$$ and hence $$s(x)=\sqrt{x}=\exp(\log\sqrt{x})$$ is elementary too. Perhaps $\arctan$ is a bit harder to deduce from the other steps in the construction of $E$, first note that \begin{eqnarray} \tan x=\frac{\sin x}{\cos x}= \frac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}= -i\frac{e^{2ix}-1}{e^{2ix}+1}=\\ -i\frac{e^{2ix}+1 -2}{e^{2ix}+1} = -i\frac{e^{2ix}+1 }{e^{2ix}+1}-i\frac{-2}{e^{2ix}+1}=\\-i +i\frac{2}{e^{2ix}+1} \end{eqnarray}

Hence if we solve for, $e^{2ix}$ in the above identity we get \begin{eqnarray} e^{2ix}=\frac{2}{1-i\tan x}-1= \frac{2-(1-i\tan x)}{1-i\tan x}= \frac{1+i\tan x}{1-i\tan x} \end{eqnarray} and taking the complex logarithm (more precisely the principal branch of the logarithm) we get $$x=\frac{1}{2i}\log\frac{1+i\tan x}{1-i\tan x}$$ That is $$t\mapsto \arctan t =\frac{1}{2i}\log\frac{1+it}{1-it}$$ belongs to $E$.

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@ZsbánAmbrus Could you please explain what you mean? What you look at is an equation - not a function. –  AD. Mar 9 '12 at 9:28
This nice, concise definition of $E$ falls short of explaining why we consider its elements "elementary". Note that polynomials are merely finite applications of addition and multiplication; $n$-th roots undo "powers" (repeated multiplication). Exponentials are merely powers with a different focus (base held constant rather than exponent); logs undo those. Trigs get into the club via ties to the complex exponential. Perhaps what makes elementary functions "elementary", then, is that they're fundamentally "arithmetical". –  Blue Mar 9 '12 at 11:19
I don't think you can include function inversion in your list. For one, the inverse of $x \mapsto xe^x$ is not usually considered elementary. For another, @Zsbán's comment might be referring to the inverse of $x \mapsto b = \sin x - ax$. –  Rahul Mar 9 '12 at 13:04
Correct. Do not include function inversion. Instead of writing down what you remember, why not cite a definitive reference? –  GEdgar Mar 9 '12 at 15:58
A minor note: it would probably be good to explain how $\sqrt{x}$ is an elementary function under your definition, since that's a bit non-trivial (you have to go through exp and log to get it). –  Steven Stadnicki Feb 28 '13 at 18:38

The original truly elementary functions are rational functions built up with the basic arithmetic operations.

Arctan and log are necessary and sufficient to integrate rational functions while sin/cos and exp are the solutions of the simplest differential equations and serve as a basis for the simplest interesting class of differential equations.

So, in a way, trigonometric functions, log and exp are the next thing found if one adds calculus to basic arithmetic.

But of course, it remains always slightly arbitrary since it is hard to justify the exclusion of elliptic functions, for example.

Yet, few people would argue that the elementary functions aren't simpler/satisfy simpler relations than elliptic functions in most respects.

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"At one time... every young mathematician was familiar with $\mathrm{sn}\,u$, $\mathrm{cn}\,u$, and $\mathrm{dn}\,u$, and algebraic identities between these functions figured in every examination" - E.H. Neville –  Guess who it is. Apr 26 '12 at 13:26
The good old times. –  Phira Apr 26 '12 at 13:30
I wonder if part of the rationale behind the exclusion of the elliptic functions is that they're inherently two-argument functions, whereas all of the elementary functions are functions of a single variable. They're also arguably more self-contained than any of the elementary functions - one is less likely to accidentally 'trip over' an elliptic function while doing other mathematical work than with any of the core elementary functions. –  Steven Stadnicki Feb 28 '13 at 21:21

There is an article an the January edition of the Notices entitled "Closed Forms: What They Are and Why We Care". http://www.ams.org/notices/201301/index.html

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This is a sort of conjectural answer.

I suspect the elementary functions are called that simply because they are so simple. This sounds a bit tautological, so let me go on. What classifies as an elementary function? I think everyone would agree that polynomials, the nth-root functions, logarithms, exponentials, sines, and cosines are all elementary functions. A couple cool things about these functions are that we completely understand their graphs, continuity and rates of growth, and all derivatives and antiderivatives of all orders more or less. And we've seen polynomials and nth-root functions since elementary school; exponentials, logs, sines and cosines since mid-secondary school. One might argue that we don't really look at exponentials or logs until we prep for calculus - that's a fair statement, perhaps.

What I'm trying to say is that we've seen these all before and get them. When I think of a non-standard function, the first that come to mind are either the logarithmic integral $li(x) = \int \frac{1}{\ln x}$ and the sine integral $si(x) = \int \frac{\sin x}{x}$. I don't know why, really. But I have to really think before I know what the sine integral looks like (the logarithmic integral happens to not be so bad mentally, so it goes).

But what about other functions. Do we call them elementary? Are rational functions elementary? I might argue that a rational function is more elementary than $\sec x$, I think, because secant is weird (in my opinion). And we don't understand arbitrary antiderivatives of secants so well. To admit a weakness, I can't currently think of the antiderivative at all. (Is is $\ln|\sec x + \tan x|$? Something like that - differentiate it and find out, I suppose).

What about compositions of elementary functions. Is $\cos \sin (x)$ elementary? I hope not. But products are. Why? Again, I would say that products are easy, compositions are weird. Compositions can change domains and ranges in not-immediately-intuitive ways. I suppose products could too, but those fit my intuition better.

To give credence, natural log is a stretch for an elementary function in my opinion. It's well studied, but not well-behaved. We do understand arbitrary antiderivatives of it, but it takes a bit of work. This is a theme with natural log, just like asking how multiplying by $\ln (x)$ or $1/\ln(x)$ affects convergence (it 'almost' never does).

This makes me think of general reciprocals. $\frac{1}{x}$ is elementary. Is $\frac{1}{\ln x}$? I sort of hope not there too - it's close to the logarithmic integral.

So in short, I would say that elementary functions are those that we all like and understand without thinking too hard. But I also don't think that the exact list of elementary functions is concrete. It would be interesting to know what mathematica or W|A considers elementary, because they both certainly have an inbuilt list.

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en.wikipedia.org/wiki/Elementary_function. According to this composition of elementary functions are also treated as elementary functions, though in general we cannot believe all the claims in wikipedia completely. –  user17762 Mar 9 '12 at 5:29
The usual convention is that elementary functions are closed under arithmetic operations and composition. So yes, $1/\log x$, $\cos(\sin x)$, rational functions, etc., are elementary functions. More information is available on Wikipedia. –  Jonas Meyer Mar 9 '12 at 5:30
This is one of the few cases where I do not agree with wikipedia's claim then, I suppose. –  mixedmath Mar 9 '12 at 7:28
@mixedmath: Wikipedia's authors did not invent the convention. This is the same meaning used in the statements of theorems such as that $e^{x^2}$ and $\frac{1}{\ln(x)}$ have no elementary antiderivative. –  Jonas Meyer Mar 9 '12 at 15:07
@mixedmath The sine integral is important because, like the error integral, it actually turned up in applications, so that people tried very hard to "solve" it before they were able to even formulate what "unsolvable" means. –  Phira Apr 26 '12 at 13:18

Another point, sometimes left out of the discussion, is the setting. Something like: A meromorphic function defined on a connected domain in the complex plane is called an elementary function if ...

Then you can do variants. Functions defined on an interval in the real line, perhaps omitting trig functions in your definition ... An abstract extension of a differential field $F(x)$ ...

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I believe that a system of axioms for elementary functions is not only hard to build, but also almost useless. Actually, there functions that some teacher/professor would call elementary and other teacher/professors would definitely not. For instance, I strongly suggest that characteristic functions of finite unions of intervals should be elementary: their graphs are simply horizontal segments, which is rather elementary. Moreover, students grow up thinking that every function is actually a $C^\infty$ function, since they believe that only "elementary" functions exist! It is time to allow elementary functions to be (at least) discontinuous at a finite numer of points.

More conservative collegues found a useful recipe: elementary functions are

1) polynomials; 2) exponentials and logarithms; 3) goniometric functions (sin, cos, arcsin, arccos, and their sons/daughters like tan, arctan, cotan, ecc.).

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Elementary functions are the functions built from the three binary arithmetic operations, addition $x+y$, multiplication $x\cdot y$ and exponentiation $x\text{^}y$ and their respective inverse operations, subtraction, division, finding a root and finding a logarithm.

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There is few to add to the preceeding answers, showing that the conventional boundary between the so called "elementary" and "special" functions is still a matter for discussion. Some papers were cited as references. In addition, a review for general public is published on Scribd (pp.18-26) : http://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales-

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Elementary functions can all be generated through an infinite sum $\sum a_n x^n$, aswell as finite number of compositions, $f((g(x))$, they are the additive completion of polynomials, they are elementary because polynomials are easy to handle.

Non-elementary functions such as the zeta-function and gamma-function, can not be written as infinite polynomials, but are most natural defined by infinite products. They are non-elementary because multiplication is more complex then addition, they are a step up in the hiearchy.

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There is am immense number of functions which do have a development as an infinite sum of the form you mention yet which are generally not said to be elementary... –  Mariano Suárez-Alvarez Mar 9 '12 at 6:05
Every differentiable function of a complex variable! If you want one with a name.... say, the Airy functions. –  Mariano Suárez-Alvarez Mar 9 '12 at 6:08
I don't know where you looked. But the Airy functions $Ai$, for example, have Taylor series developments around zero with real coefficients and valid on the whole of $\mathbb R$. –  Mariano Suárez-Alvarez Mar 9 '12 at 6:20
I think you have described all analytic functions, which is a strictly larger set. The error function, for example, is not elementary but its Taylor series converges everywhere. –  Rahul Mar 9 '12 at 6:39
By golly, how did I know you were going to ask that. –  anon Mar 9 '12 at 6:59