The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece and India. In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote: 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 hyperbolaxy = 1 by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola. The area under the hyperbola was shown to have the scaling property of constant area for a constant ratio of bounds. The natural logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. By introducing these transcendental functions and noting the bijection property that implies an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function. The exponential function is written Euler identified it with the infinite series where k! denotes the factorial of k. The even and odd terms of this series provide sums denoting cosh x and sinh x, so that These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing 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.
Examples
The following functions are transcendental: In particular, for ƒ2 if we set c equal toe, the base of the natural logarithm, then we get that ex is a transcendental function. Similarly, if we set c equal to e in ƒ5, then we get that is a transcendental function.
Most familiar transcendental functions, including the special functions of mathematical physics, are solutions of algebraic differential equations. Those that are not, such as the gamma and the zeta functions, are called transcendentally transcendental or hypertranscendental functions.
Exceptional set
If is an algebraic function and is an algebraic number then is also an algebraic number. The converse is not true: there are entire transcendental functions such that is an algebraic number for any algebraic For a given transcendental function the set of algebraic numbers giving algebraic results is called the exceptional set of that function. Formally it is defined by: In many instances the exceptional set is fairly small. For example, this was proved by Lindemann in 1882. In particular exp = e is transcendental. Also, since exp = −1 is algebraic we know that iπ cannot be algebraic. 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. Here are some other known exceptional sets:
A consequence of Schanuel's conjecture in transcendental number theory would be that
A function with empty exceptional set that does not require assuming Schanuel's conjecture is
While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraic numbers, say A, there is a transcendental function whose exceptional set is A. The subset does not need to be proper, meaning that A can be the set of algebraic numbers. This directly implies that there exist transcendental functions that produce transcendental numbers only when given transcendental 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.
Dimensional analysis
In dimensional analysis, transcendental functions are notable because they make sense only when their argument is dimensionless. Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, log is a nonsensical expression, unlike log or log meters. One could attempt to apply a logarithmic identity to get log + log, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.
