Editing Transcendental function (section)

==Definition==
Formally, an [[analytic function]] <math>f</math> of one [[real number|real]] or [[complex number|complex]] variable is '''transcendental''' if it is [[algebraically independent]] of that variable.<ref>{{cite book |first=M. |last=Waldschmidt |title=Diophantine approximation on linear algebraic groups |publisher=Springer |location= |date=2000 |isbn=978-3-662-11569-5 |pages= |url={{GBurl|Wrj0CAAAQBAJ|pg=PR9}}}}</ref> This means the function does not satisfy any polynomial equation. For example, the function <math>f</math> given by

:<math>f(x)=\frac{ax+b}{cx+d}</math> for all <math>x</math>

is not transcendental, but algebraic, because it satisfies the polynomial equation

:<math>(ax+b)-(cx+d)f(x)=0</math>.

Similarly, the function <math>f</math> that satisfies the equation

:<math>f(x)^5+f(x)=x</math> for all <math>x</math>

is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations.

This definition can be extended to [[Function of several real variables|functions of several variables]].