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 complex number that is not a zero of any polynomial with rational coefficients.

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) = a_n xⁿ + ... + a₁ x + a₀,

with all a_i ∈ F, such that P(ξ) = 0.

In the case of the field C of complex numbers or the field R of real numbers, a transcendental number is a number which is transcendental over the field Q of rational numbers.

See also

• Transcendental number
• Algebraic number