Algebraic function
Algebraic functions are a special class of functions that are examined in particular in the mathematical sub-area of algebra . They are the solution to an algebraic equation . Functions that are not algebraic are called transcendent functions .
The theory of algebraic functions was developed in the past from the three mathematical sub-areas of function theory , arithmetic algebraic geometry and algebraic geometry .
definition
A function in variables is called an algebraic function if there is an irreducible polynomial in variables and coefficients in a field such that the algebraic equation
solves.
So a function of a variable is algebraic if it has the equation
fulfilled, where polynomials are in the variable .
properties
- Since the definition required that the polynomials are irreducible, it can be proven that for every algebraic function there is exactly one irreducible polynomial with the exception of one constant . The degree of the polynomial in the variable is then called the degree of the algebraic function.
- For degrees , all algebraic functions can be represented as rational functions and for degrees , and they can all be represented as the square or cube roots of a rational function. In general, this is not possible for grades .
- Algebraic functions of a variable over the field of complex numbers are meromorphic .
Examples
- Power functions with rational exponents .
- Polynomial functions
- Rational functions or broken-rational functions
- Root function
Transcendent functions
A function is called transcendent if it is not algebraic. These include, for example
- the exponential function
- the logarithm function
- Circle and hyperbolic functions
Web links
- Eric W. Weisstein : Algebraic Function . In: MathWorld (English).
- Zhizhchenko: Algebraic function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
Individual evidence
- ^ Josef Naas , Hermann Ludwig Schmid : Mathematical dictionary. With the inclusion of theoretical physics. Volume 1: A - K. 3rd edition, unchanged reprint. Akademie-Verlag et al., Berlin et al. 1979, ISBN 3-519-02400-4 .