Transcendence (mathematics)
In mathematics, a transcendental function is a function which is not expressible as a composition of a finite number of elementary operations, or inverses of functions so constructible, where the elementary operations consist of addition, multiplication, taking additive or multiplicative inverses, and integer root extraction. Transcendental functions include all the trigonometric functions and logarithmic functions, along with most other special functions in mathematics.
A transcendental number is a real number that is not a zero of any non-zero polynomial with rational coefficients.
This latter definition can be generalized as follows. A transcendental element ξ of a field extension K over the field F is an element that is not the solution of a polynomial equation with coefficients in F, i.e., if there exists no polynomial
- P(x) = an xn + ... + a1 x + a0,
with all ai ∈ F, such that P(ξ) = 0.
An algebraic function f(x) is a solution of an equation P(f(x)) = 0, where P is a polynomial. A transendental function is a function that is neither a polynomial function nor an algebraic function.