In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable that can be written using only the basic operations of addition, subtraction, multiplication, and division (without the need of taking limits).
[3] This means the function does not satisfy any polynomial equation.
given by is not transcendental, but algebraic, because it satisfies the polynomial equation Similarly, the function
that satisfies the equation is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations.
The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya).
In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: The mathematical notion of continuity as an explicit concept is unknown to Ptolemy.
That he, in fact, treats these functions as continuous appears from his unspoken presumption that it is possible to determine a value of the dependent variable corresponding to any value of the independent variable by the simple process of linear interpolation.
[4]A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard Euler in 1748 in his Introduction to the Analysis of the Infinite.
These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola xy = 1 by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola.
These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing (−1)k into the series, resulting in alternating series.
After Euler, mathematicians view the sine and cosine this way to relate the transcendence to logarithm and exponent functions, often through Euler's formula in complex number arithmetic.
can be replaced by any other irrational number, and the function will remain transcendental.
can be replaced by any other positive real number base not equaling 1, and the functions will remain transcendental.
The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses of all of these.
Transcendental functions cannot be defined using only the operations of addition, subtraction, multiplication, division, and
The indefinite integral of many algebraic functions is transcendental.
Similarly, the limit or the infinite sum of many algebraic function sequences is transcendental.
turns out to equal the hyperbolic cosine function
In fact, it is impossible to define any transcendental function in terms of algebraic functions without using some such "limiting procedure" (integrals, sequential limits, and infinite sums are just a few).
Differential algebra examines how integration frequently creates functions that are algebraically independent of some class, such as when one takes polynomials with trigonometric functions as variables.
Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations.
[6] For a given transcendental function the set of algebraic numbers giving algebraic results is called the exceptional set of that function.
In many instances the exceptional set is fairly small.
Since i is algebraic this implies that π is a transcendental number.
In general, finding the exceptional set of a function is a difficult problem, but if it can be calculated then it can often lead to results in transcendental number theory.
[10] The subset does not need to be proper, meaning that A can be the set of algebraic numbers.
Alex Wilkie also proved that there exist transcendental functions for which first-order-logic proofs about their transcendence do not exist by providing an exemplary analytic function.
[11] In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless (possibly after algebraic reduction).
Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
One could attempt to apply a logarithmic identity to get log(5) + log(metres), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.