Transcendental function

A transcendental function is a function that is not algebraic. Such a function cannot be expressed as a solution of a polynomial equation whose coefficients are themselves polynomials with rational coefficients.[1] Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.

Definition

Formally, an analytic function ƒ(z) of the real or complex variables z1,…,zn is transcendental if z1, …, zn, ƒ(z) are algebraically independent,[2] i.e., if ƒ is transcendental over the field C(z1, …,zn).

A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.

Examples

The following functions are transcendental:

$f_1(x) = x^\pi \$
$f_2(x) = c^x, \ c \ne 0, 1$
$f_3(x) = x^{x} = {{^2}x} \$
$f_4(x) = x^{\frac{1}{x}} \$
$f_5(x) = \log_c x, \ c \ne 0, 1$
$f_6(x) = \sin{x}$

Note that in particular for ƒ2 if we set c equal to e, 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 ln(x), the natural logarithm, is a transcendental function. For more information on the second notation of ƒ3, see tetration.

Algebraic and transcendental functions

The logarithm and the exponential function are examples of transcendental functions. Transcendental function is a term often used to describe the trigonometric functions (sine, cosine, tangent, their reciprocals cotangent, secant, and cosecant, the now little-used versine, haversine, and coversine, their analogs the hyperbolic functions and so forth).

A function that is not transcendental is said to be algebraic. Examples of algebraic functions are rational functions and the square root function.

The operation of taking the indefinite integral of an algebraic function is a source of transcendental functions. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector. Thus the hyperbolic angle and the hyperbolic functions sinh, cosh, and tanh are all transcendental.

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.

Dimensional analysis

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. For example, log(5 meters) is a nonsensical expression, unlike log(5 meters / 3 meters)  or  log(3) meters. One could attempt to apply a logarithmic identity to get log(10) + log(m), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

Exceptional set

If ƒ(z) is an algebraic function and α is an algebraic number then ƒ(α) will also be an algebraic number. The converse is not true: there are entire transcendental functions ƒ(z) such that ƒ(α) is an algebraic number for any algebraic α. In many instances, however, the set of algebraic numbers α where ƒ(α) is algebraic is fairly small. For example, if ƒ is the exponential function, ƒ(z) = ez, then the only algebraic number α where ƒ(α) is also algebraic is α = 0, where ƒ(α) = 1. For a given transcendental function this set of algebraic numbers giving algebraic results is called the exceptional set of the function,[3][4] that is the set

$\mathcal{E}(f)=\{\alpha\in\overline{\mathbf{Q}}\,:\,f(\alpha)\in\overline{\mathbf{Q}}\}.$

If this set can be calculated then it can often lead to results in transcendence theory. For example, Lindemann proved in 1882 that the exceptional set of the exponential function is just {0}. In particular exp(1) = e is transcendental. Also, since exp(iπ) = -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 it has been calculated for some functions:

• $\mathcal{E}(\exp)=\{0\}$,
• $\mathcal{E}(2^{x})=\mathbf{Q}$,
• This result is a corollary of the Gelfond–Schneider theorem which says that if α is algebraic and not 0 or 1, and if β is algebraic and irrational then αβ is transcendental. Thus the function 2x could be replaced by cx for any algebraic c not equal to 0 or 1. Indeed, we have:
• $\mathcal{E}(x^x)=\mathcal{E}(x^{\frac{1}{x}})=\mathbf{Q}\setminus\{0\}.$
• A function with empty exceptional set that doesn't require one to assume this conjecture is the function ƒ(x) = exp(1 + πx).

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 ƒ whose exceptional set is A.[6] Since, as mentioned above, this includes taking A to be the whole set of algebraic numbers, there is no way to determine if a function is transcendental just by looking at its values at algebraic numbers. In fact, Alex Wilkie showed that the situation is even worse: he constructed a transcendental function ƒ: RR that is analytic everywhere but whose transcendence cannot be detected by any first-order method.[7]