Elementary function: Difference between revisions
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The '''Elementary Functions''' are the most basic [[function|functions]] arising in the study of [[calculus]]. They include the [[polynomial|polynomials]], which are the object of study of [[elementary algebra]]. More generally they include all of the [[algebraic function|algebraic functions]] as well as the most basic [[transcendental functions]]: the [[exponential function]], the [[logarithm]], the [[trigonometric function|trigonometric functions]], and the [[hyperbolic function|hyperbolic functions]]. Furthermore, finite combinations of the previous functions | The '''Elementary Functions''' are the most basic [[function|functions]] arising in the study of [[calculus]]. They include the [[polynomial|polynomials]], which are the object of study of [[elementary algebra]]. More generally they include all of the [[algebraic function|algebraic functions]] as well as the most basic [[transcendental functions]]: the [[exponential function]], the [[logarithm]], the [[trigonometric function|trigonometric functions]], and the [[hyperbolic function|hyperbolic functions]]. Furthermore, finite combinations of the previous functions, the four elementary operations of [[addition]], [[subtraction]], [[multiplication]], and [[division]], and [[composition|function composition]] are also elementary functions. | ||
==Overview== | ==Overview== | ||
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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. | 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 | ===Algebraic functions=== | ||
====Polynomials and | ====Polynomials and rational functions==== | ||
In a sense, the [[identity function]] <math>I(x) = x </math> is the most elementary function. | In a sense, the '''[[identity function]]''' <math>I(x) = x </math> is the most elementary function. We can also consider the '''[[constant function|constant functions]]''' to be elementary. These are the functions of the form <math>f(x) = c</math> for some number <math>c</math>. From the constant functions and the identity function, we can construct the polynomials. | ||
The '''[[monomial]]''' <math>ax^n</math> is the identity function multiplied with itself <math>n</math> times and then multiplied by the [[constant]] <math>a</math>. The '''[[polynomial|polynomials]]''' then are [[finite]] [[linear combination|linear combinations]] of the monomials and the constant functions. | |||
From the polynomials, it is possible to construct more elementary functions: the '''[[rational function|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 <math>r(x)</math> that can be written as | |||
:<math>r(x) = \frac{p(x)}{q(x)} </math> | :<math>r(x) = \frac{p(x)}{q(x)} </math> | ||
where <math>p</math> and <math>q</math> are polynomials. | where <math>p(x)</math> and <math>q(x)</math> are polynomials. | ||
====In | ====In general==== | ||
The rational functions are a [[set|subset]] of the [[algebraic function|algebraic functions]]. They are characterized in general as follows. | The rational functions are a [[set|subset]] of the '''[[algebraic function|algebraic functions]]'''. They are characterized in general as follows. | ||
Consider the polynomials in two variables. If <math>p(x,y)</math> is such a polynomial, then the [[set]] of [[point|points]] that satisfy <math>p(x,y) = 0</math> 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]] | Consider the polynomials in two variables. If <math>p(x,y)</math> is such a polynomial, then the [[set]] of [[point|points]] that satisfy <math>p(x,y) = 0</math> 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]] | ||
:<math>x^{2} + y^{2} = 0</math> | :<math>x^{2} + y^{2} - 1= 0</math> | ||
Another example is the curve whose points satisfy | Another example is the curve whose points satisfy | ||
:<math>y^{3} | :<math>y^{3} + y - x = 0</math> | ||
In general, algebraic curves are not the [[graph|graphs]] of functions. The first example is not the graph of a function in either <math>x</math> or <math>y</math>. The second example is the graph of a function of <math>x</math>. Since it passes the [[ | In general, algebraic curves are not the [[graph|graphs]] of functions. The first example is not the graph of a function in either <math>x</math> or <math>y</math>. The second example is the graph of a function of <math>x</math>. Since it passes the [[horizontal line test]], it is also the graph of a function of <math>y</math>. 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 <math>f(x) = x^{3} | 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 <math>f(x) = x^{3} + x</math> is also an algebraic function. [[Root|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 | ===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 mathematics|pure]] and [[applied mathematics]] and also because they have many desirable and interesting properties that make them special. | 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 mathematics|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 | It turns out that there are really only two elementary transcendental functions and the remainder can be formed from them. | ||
====Exponential | ====Exponential function==== | ||
The most important of all transcendental functions is the exponential function defined as: | The most important of all transcendental functions is the '''[[exponential function]]''' defined as: | ||
:<math>\exp(x) = e^{x}</math> | :<math>\exp(x) = e^{x}</math> | ||
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====Logarithm==== | ====Logarithm==== | ||
The second transcendental that is considered elementary is the [[inverse]] of the exponential function, the logarithm. The logarithm is denoted <math>\ln(x)</math>. It is the unique function that satisfies the [[equation]]: | The second transcendental that is considered elementary is the [[inverse]] of the exponential function, the '''[[logarithm]]'''. The logarithm is denoted <math>\ln(x)</math>. It is the unique function that satisfies the [[equation]]: | ||
:<math>\exp({\ln(x)}) = x</math> | :<math>\exp({\ln(x)}) = x</math> | ||
====Exponential and | ====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 <math>f(x) = a^{x}</math> and their inverses be considered elementary for arbitrary <math>a</math>? The answer is because any such function is equal to <math>\exp({\ln(a)*x}) = e^{\ln(a)x}</math> and so we have already accounted for these functions. | 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 <math>f(x) = a^{x}</math> and their inverses be considered elementary for arbitrary <math>a</math>? The answer is because any such function is equal to <math>\exp({\ln(a)*x}) = e^{\ln(a)x}</math> and so we have already accounted for these functions. | ||
====Trigonometric | ====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 <math>sin(x)</math> and <math>cos(x)</math>. The other trigonometric functions can be constructed using sine, cosine, and the elementary operations. | 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 <math>sin(x)</math> and <math>cos(x)</math>. 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]]. | 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 | and | ||
:<math>cos(x) = \frac{e^{ | :<math>cos(x) = \frac{e^{ix}+e^{-ix}}{2}</math> | ||
Where <math>i</math> is the [[imaginary number|imaginary unit]]. | Where <math>i</math> is the [[imaginary number|imaginary unit]]. | ||
====Hyperbolic | ====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 (<math>sinh(x)</math> and <math>cosh(x)</math>, respectively). These two functions are built from the exponential function in a way analogous to sine and cosine. | 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]]''' (<math>sinh(x)</math> and <math>cosh(x)</math>, respectively). These two functions are built from the exponential function in a way analogous to sine and cosine. | ||
:<math>sinh(x) = \frac{e^{x}–-e^{-x}}{2}</math> | :<math>sinh(x) = \frac{e^{x}–-e^{-x}}{2}</math> | ||
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:<math>cosh(x) = \frac{e^{x}+e^{-x}}{2}</math> | :<math>cosh(x) = \frac{e^{x}+e^{-x}}{2}</math> | ||
Like the trigonometric functions, the hyperbolic functions have a geometric significance. sinh and cosh are to a [[hyperbola]] what sin and cos are to a [[circle]]. This is the source of their name. | Like the trigonometric functions, the hyperbolic functions have a geometric significance. <math>sinh(x)</math> and <math>cosh(x)</math> are to a [[hyperbola]] what <math>sin(x)</math> and <math>cos(x)</math> 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 | ==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, | The [[derivative]] of an elementary function is also an elementary function. However, the [[antiderivative]] of an elementary function is not necessarily elementary. For example, | ||
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==References== | ==References== | ||
G. H. Hardy, A Course Of Pure Mathematics, 10th ed., Cambridge University Press, 1908, 1952. | G. H. Hardy, A Course Of Pure Mathematics, 10th ed., Cambridge University Press, 1908, 1952.[[Category:Suggestion Bot Tag]] |
Latest revision as of 06:01, 11 August 2024
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 is the most elementary function. We can also consider the constant functions to be elementary. These are the functions of the form for some number . From the constant functions and the identity function, we can construct the polynomials.
The monomial is the identity function multiplied with itself times and then multiplied by the constant . 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 that can be written as
where and 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 is such a polynomial, then the set of points that satisfy 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
Another example is the curve whose points satisfy
In general, algebraic curves are not the graphs of functions. The first example is not the graph of a function in either or . The second example is the graph of a function of . Since it passes the horizontal line test, it is also the graph of a function of . 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 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:
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
Logarithm
The second transcendental that is considered elementary is the inverse of the exponential function, the logarithm. The logarithm is denoted . It is the unique function that satisfies the equation:
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 and their inverses be considered elementary for arbitrary ? The answer is because any such function is equal to 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 and . 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.
- Failed to parse (syntax error): {\displaystyle sin(x) = \frac{e^{ix}-–e^{-ix}}{2i}}
and
Where 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 ( and , respectively). These two functions are built from the exponential function in a way analogous to sine and cosine.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sinh(x) = \frac{e^{x}–-e^{-x}}{2}}
Like the trigonometric functions, the hyperbolic functions have a geometric significance. and are to a hyperbola what and 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,
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.