Algebraic Function(redirected from Algebraic functions)
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algebraic function[¦al·jə¦brā·ik ′fəŋk·shən]
a function that satisfies an algebraic equation; one of the most important functions studied in mathematics. Among the algebraic functions, polynomials and quotient polynomials—for example, and (1 + x + x2)/(2 + x3)—are called rational and all the rest are called irrational. The simplest examples of the latter are algebraic functions expressed by means of radicals—for example, and However, there are certain algebraic functions that cannot be expressed with radicals—the function y = f (x) which satisfies the equation y5 + 5yx4 + 5x5 = 0. Examples of nonalgebraic, or so-called transcendental functions, encountered in school algebra courses, are the power function xα, if α is an irrational number; the exponential function ax; the logarithmic function; and so forth. The general theory of algebraic functions is an extensive mathematical discipline having important connections with the theory of analytic functions (of which algebraic functions constitute a special class), algebra, and algebraic geometry. The most general algebraic function of many variables, u = f(x, y, z., ...), is defined as a function satisfying an equation of the type
P0(x, y, z, . . . )un + P1 (x, y, z, . . .)un−1
(1) + . . . + Pn (x, y, z, . . . ) = 0
where P0,P1,. . . Pn are any polynomials with respect to x, y, z, . . . . The entire expression in the left member represents a certain polynomial with respect to x, y, z, . . . and u. It may be considered irreducible—that is, not factorable into polynomials of lower degree; in addition, polynomial P0 can be considered as not identically equal to zero. If n = 1, then u represents a rational function (u = −P1/P0), a partial case of which—the integral rational function—is the polynomial (if P0 = const ≠ 0). When n > 1, we have an irrational function; when n = 2, the function is expressed in terms of polynomials with the use of a square root; when n = 3 or 4, for u we have an expression containing both square and cube roots.
When n ≥ 5, the irrational function u can no longer be expressed in the general case, in terms of a finite number of any roots of polynomials. An irrational algebraic function is always many-valued; it is, precisely, for our given designations and assumptions, an n-valued analytic function of variables x, y, z,.....