algebraic function
A function of one variable is said to be algebraic if it satisfies a polynomial equation whose coefficients are polynomials in the same variable. Namely, the function f(x) is algebraic if y=f(x) is a solution of an equation of the form
pn(x)yn+⋯+p1(x)y+p0(x)=0, |
where the p0(x),p1(x),…,pn(x) are polynomials in x. A function that satisfies no such equation is said to be transcendental.
The graph of an algebraic function is an algebraic curve, which is, loosely speaking, the zero set of a polynomial in two variables.
Examples
Any rational function f(x)=P(x)/Q(x) is algebraic, since y=f(x) is a solution to Q(x)y-P(x)=0.
The function f(x)=√x is algebraic, since y=f(x) is a solution to y2-x=0. The same is true for any power function xn/m, with n and m integers, it satisfies the equation ym-xn=0.
It is known that the functions ex and lnx are transcendental. Many special functions, such as Bessel functions, elliptic integrals, and others are known to be transcendental.
Remark. There is also a version of an algebraic function defined on algebraic systems. Given an algebraic system A, an n-ary algebraic function on A is an n-ary operator f(x1,…,xn) on A such that there is an (n+m)-ary polynomial (http://planetmath.org/PolynomialsInAlgebraicSystems) p(x1,…,xn,xn+1,…,xn+m) on A for some non-negative integer m, and elements a1,…,am∈A such that
f(x1,…,xn)=p(x1,…,xn,a1,…,am). |
Equivalently, given an algebraic system A, if we associate each element a of A a corresponding symbol, also written a, we may form an algebraic system A′ from A by adjoining every symbol a to the type of A considered as a unary operator symbol, and leaving everything else the same. Then an algebraic function on A is just a polynomial on A′ (and vice versa).
For example, in a ring R, a function f on R given by f(x)=anxn+⋯+a1x+a0 where ai∈R is a unary algebraic function on R, as f(x)=p(x,a0,…,an), where p is an (n+2)-ary polynomial on R given by p(x,x0,…,xn)=xnxn+⋯+x1x+x0.
References
- 1 G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).
- 2 S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).
Title | algebraic function |
Canonical name | AlgebraicFunction |
Date of creation | 2013-03-22 15:19:24 |
Last modified on | 2013-03-22 15:19:24 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 15 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 08A40 |
Classification | msc 26A09 |
Related topic | ElementaryFunction |
Related topic | PropertiesOfEntireFunctions |
Related topic | PolynomialsInAlgebraicSystems |
Defines | transcendental function |
Defines | transcendental |
Defines | algebraic curve |