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In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of ...
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients. Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
Following an initial solution, iteration of the procedure may generate additional terms of an asymptotic expansion providing a more accurate solution. [1] [2] An early example of the dominant balance method is the Newton polygon method. Newton developed this method to find an explicit approximation for an algebraic function.
Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
This algebraic function can be written as the right side of the solution equation y = f(x). Written like this, f is a multi-valued implicit function. Algebraic functions play an important role in mathematical analysis and algebraic geometry. A simple example of an algebraic function is given by the left side of the unit circle equation:
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