Elementary function

The Elementary Functions are the most basic functions arising in the study of calculus. They include the polynomials, which are the object of study of elementary algebra. More generally they include all of the algebraic functions as well as the most basic transcendental functions: the exponential function, the logarithm, the trigonometric functions, and the hyperbolic functions. Furthermore, finite combinations of the previous functions, the four elementary operations of addition, subtraction, multiplication, and division, and function composition are also elementary functions.

Overview

The elementary functions can be divided into two groups: The functions that arise from polynomials, and the functions that don't. The former are simple to construct whereas the latter must be specified by their special properties.

Algebraic Functions

Polynomials and Rational Functions

In a sense, the identity function $I(x)=x$ is the most elementary function. We can also consider the constant functions to be elementary. These are the functions of the form $f(x)=c$ for some number $c$. From the constant functions and the identity function, we can construct the polynomials.

The monomial $ax^{n}$ is the identity function multiplied with itself $n$ times and then multiplied by the constant $a$. The polynomials then are finite linear combinations of the monomials and the constant functions.

From the polynomials, it is possible to construct more elementary functions: the rational functions. The product of two polynomials is again a polynomial. However, the quotient of two polynomials need not be one. A function that is the quotient of two polynomials is called a rational function. In other words, a rational function is a function $r(x)$ that can be written as

$r(x)={\frac {p(x)}{q(x)}}$

where $p(x)$ and $q(x)$ are polynomials.

In General

The rational functions are a subset of the algebraic functions. They are characterized in general as follows.

Consider the polynomials in two variables. If $p(x,y)$ is such a polynomial, then the set of points that satisfy $p(x,y)=0$ form a curve in the plane. Such curves are called algebraic curves. For example, the circle is such a curve. It is the set of solutions to the algebraic equation

$x^{2}+y^{2}-1=0$

Another example is the curve whose points satisfy

$y^{3}+y-x=0$

In general, algebraic curves are not the graphs of functions. The first example is not the graph of a function in either $x$ or $y$. The second example is the graph of a function of $x$. Since it passes the horizontal line test, it is also the graph of a function of $y$. Although not all algebraic curves are the graphs of functions, there is a theorem (the implicit function theorem) that tells us that under certain circumstances* we can restrict our attention to a subset of the curve around a point and get a curve that is the graph of a function.

Any function whose graph can be derived from an algebraic curve in this way is called an algebraic function. The polynomials and rational functions are algebraic functions. The inverse of the function $f(x)=x^{3}+x$ is also an algebraic function. Roots, such as the Square Root are also algebraic because they are the inverses of certain polynomials. Finally, the sum, difference, product, quotient, and composition of any two algebraic functions is also algebraic. More generally, the finite combination of the elementary operations and composition of the algebraic functions is also an algebraic function.

Transcendental Functions

If a function is not algebraic, it is called a transcendental function. There are some such functions that are considered elementary. They are included because they are ubiquitous in both pure and applied mathematics and also because they have many desirable and interesting properties that make them special.

It turns out that there are really only two elementary transcendental functions and the remainder can be formed from them.

Exponential Function

The most important of all transcendental functions is the exponential function defined as:

$\exp(x)=e^{x}$

Where e is Euler's number. It has many beautiful and remarkable properties that make it a natural augmentation to our current set of elementary functions. For example, it satisfies the differential equation

${\frac {df(x)}{dx}}=f(x)$

Logarithm

The second transcendental that is considered elementary is the inverse of the exponential function, the logarithm. The logarithm is denoted $\ln(x)$. It is the unique function that satisfies the equation:

$\exp({\ln(x)})=x$

Exponential and Logarithm Functions in General

It may seem that we can generalize the exponential function to get more functions that we should consider elementary. Why shouldn't functions of the form $f(x)=a^{x}$ and their inverses be considered elementary for arbitrary $a$? The answer is because any such function is equal to $\exp({\ln(a)*x})=e^{\ln(a)x}$ and so we have already accounted for these functions.

Trigonometric Functions

Because the trigonometric functions are basic to geometry and applied mathematics, they are also considered elementary. The two most basic trigonometric functions are sine and cosine, denoted respectively as $sin(x)$ and $cos(x)$. The other trigonometric functions can be constructed using sine, cosine, and the elementary operations.

It is a remarkable thing that the sine and cosine functions can be defined from the exponential function when it is defined over the complex numbers.

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and

$cos(x)={\frac {e^{ix}+e^{-ix}}{2}}$

Where $i$ is the imaginary unit.

Hyperbolic Functions

The final set of functions that completes our repertoire of elementary functions is the set of hyperbolic functions. Just as the trigonometric functions are built up from sine and cosine, the hyperbolic functions are built up from hyperbolic sine and hyperbolic cosine ($sinh(x)$ and $cosh(x)$, respectively). These two functions are built from the exponential function in a way analogous to sine and cosine.

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$cosh(x)={\frac {e^{x}+e^{-x}}{2}}$

Like the trigonometric functions, the hyperbolic functions have a geometric significance. $sinh(x)$ and $cosh(x)$ are to a hyperbola what $sin(x)$ and $cos(x)$ are to a circle. This is the source of their name. The hyperbolic functions also play a significant role in applied mathematics, particularly in the discipline of engineering.

Non-elementary Functions

The derivative of an elementary function is also an elementary function. However, the antiderivative of an elementary function is not necessarily elementary. For example,

$\int e^{x^{2}}\,dx$

is not elementary.

There are many functions that are very useful but are not elementary. In general, any function that is important enough to be given a name is called a special function. Often, special functions are solutions to differential equations or integral equations of elementary functions. Some examples of special functions are the error function, the Riemann Zeta Function, the Bessel functions, and the Gamma Function.

References

G. H. Hardy, A Course Of Pure Mathematics, 10th ed., Cambridge University Press, 1908, 1952.