algebraic function

Definitions

Informally, any function expressible using (only) the operations of addition, subtraction, multiplication, division and raising to a rational power; more precisely, any continuous function definable as the root of some polynomial equation.

Equivalents

Examples

“Quite often, algebraic functions are algebraic expressions using a finite number⟳ of terms and involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Some algebraic functions, however, cannot be expressed by such finite expressions.”

“The value of an algebraic function at an algebraic number⟳ is always an algebraic number⟳.”

“1989, Manuel Bronstein, An Algorithm for the Integration of Elementary Functions, James H. Davenport (editor), EUROCAL '87: European Conference on Computer Algebra, Proceedings, Springer, LNCS 378, page 491, Trager (1984) recently gave a new algorithm for the indefinite integration of algebraic functions.”

“1992, A. A. Goldberg, V. A. Pyana, Uniqueness Theorems for Algebraic Functions, A. B. Sossinsky (translation editor), Boris Yakovlevich Levin (editor), Entire and Subharmonic Functions, American Mathematical Society, page 199, It is known that the algebraic function w=w(z) enables one to determine⟳ the polynomial P(z,w) up to a constant factor.”

“1993, Goro Kato (translator), Kenkichi Iwasawa, Algebraic Functions, American Mathematical Society, page xiv, Riemann provided a clear picture of many difficult topics. His most significant contribution, rather than his construction of a Riemann surface from an algebraic function, was to derive the existence of algebraic functions as he built his theory based on the concept of a Riemann surface.”

CEFR level