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 ${\displaystyle y=f(x_{1},\dotsc ,x_{n})}$ ${\displaystyle n}$ ${\displaystyle P}$ ${\displaystyle n+1}$ ${\displaystyle f}$

${\displaystyle P(f(x_{1},\dotsc ,x_{n}),x_{1},\dotsc ,x_{n})=0}$

solves.

So a function of a variable is algebraic if it has the equation ${\displaystyle y=f(x)}$

${\displaystyle P_{n}(x)y^{n}+\dotsb +P_{1}(x)y+P_{0}(x)=0}$

fulfilled, where polynomials are in the variable . ${\displaystyle P_{1},\dotsc ,P_{n}}$ ${\displaystyle x}$

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. ${\displaystyle y=f(x_{1},\dotsc ,x_{n})}$ ${\displaystyle P}$ ${\displaystyle P(y,x_{1},\dotsc ,x_{n})=0}$ ${\displaystyle P}$ ${\displaystyle y}$
• 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 . ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$ ${\displaystyle k>4}$
• Algebraic functions of a variable over the field of complex numbers are meromorphic . ${\displaystyle \mathbb {C} }$

Examples

• Power functions with rational exponents . ${\displaystyle y=x^{r}}$ ${\displaystyle r}$
• 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
  • Trigonometric function
  • Hyperbolic function
  • Arc function
  • Area function

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

1. ^ 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 .