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Wikipedia / Transcendental function

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). This is in contrast to an algebraic function.

The most familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses, such as the logarithm and inverse trigonometric functions. All special functions such as the gamma, error, bessel, and Riemann zeta functions are transcendental.

Equations including transcendental functions are transcendental equations.

Definition

Formally, an analytic function of one real or complex variable is transcendental if it is algebraically independent of that variable. given by

is not transcendental, but algebraic, because it satisfies the polynomial equation

  1. .

that satisfies the equation

is not transcendental, but algebraic, even though it cannot be written as a finite expression involving the basic arithmetic operations.

This definition can be extended to functions of several variables.

History

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:

A revolutionary understanding of these circular functions occurred in the 18th 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 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 hyperbolic 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 denotes the factorial of .

The even and odd terms of this series provide sums denoting and These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing 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:

, the exponent , the base denotes the analytic extension of the factorial function via the gamma function is its reciprocal, an entire function, and the function will remain transcendental.

Algebraic and transcendental functions

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. Less familiar are the special functions of analysis, such as the gamma, elliptic, and zeta functions, all of which are transcendental. The generalized hypergeometric and Bessel functions are transcendental in general, but algebraic for some special parameter values.

is any integer), without using some "limiting process".

A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functions and the square root (see Abel–Ruffini theorem and Bring radical).

The indefinite integral. 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.

Transcendentally transcendental functions

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 number then is also an algebraic number. The converse is not true: there are entire transcendental functionss 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:

this was proved by Lindemann in 1882 (see Lindemann–Weierstrass theorem). In particular is transcendental. Also, since is algebraic we know that cannot be algebraic. Since is algebraic this implies that 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:

While calculating the exceptional set for a given function is not easy, it is known that given any subset of the algebraic numbers, say , there is a transcendental function whose exceptional set is . The subset does not need to be proper, meaning that 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 (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, is a nonsensical expression, unlike or . One could attempt to apply a logarithmic identity to get , which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

See also

References