Transcendental function

Last updated October 06, 2024

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 the basic operations of addition, subtraction, multiplication, and division. This is in contrast to an algebraic function.^[1]^[2]

Definition

Formally, an analytic function $f (z)$ of one real or complex variable $z$ is transcendental if it is algebraically independent of that variable.^[3] This means the function does not satisfy a polynomial equation. For example the function

f(x)={\frac {ax+b}{cx+d}}

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

(ax+b)-(cx+d)f(x)=0.

Similarly, the function $f(x)$ that satisfies the equation

f(x)^{5}+f(x)=x

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:

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 .

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 $\exp(x)=e^{x}$ . Euler identified it with the infinite series ${\textstyle \sum _{k=0}^{\infty }x^{k}/k!}$ , 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 $e^{x}=\cosh x+\sinh x.$ 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.

Examples

Let c be a positive constant. The following functions are transcendental:

${\begin{aligned}f_{1}(x)&=x^{\pi }\\[2pt]f_{2}(x)&=c^{x}\\[2pt]f_{3}(x)&=x^{x}\\f_{4}(x)&=x^{\frac {1}{x}}={\sqrt[{x}]{x}}\\[2pt]f_{5}(x)&=\log _{c}x\\[2pt]f_{6}(x)&=\sin {x}\end{aligned}}$

For the second function $f_{2}(x)$ , if we set $c$ equal to $e$ , the base of the natural logarithm, then we get that $e^{x}$ is a transcendental function. Similarly, if we set $c$ equal to $e$ in $f_{5}(x)$ , then we get that $f_{5}(x)=\log _{e}x=\ln x$ (that is, the natural logarithm) is a transcendental function.

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.

A function that is not transcendental is algebraic. Simple examples of algebraic functions are the rational functions and the square root function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions, as shown by the example above with $f(x)^{5}+f(x)=x$ (see Abel–Ruffini theorem).

The indefinite integral of many algebraic functions is transcendental. For example, the logarithm function arose from the reciprocal function in an effort to find the area of a hyperbolic sector.

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.^[5]

Exceptional set

If $f$ is an algebraic function and $\alpha$ is an algebraic number then $f (α)$ is also an algebraic number. The converse is not true: there are entire transcendental functions $f$ such that $f (α)$ is an algebraic number for any algebraic $α$ .^[6] For a given transcendental function the set of algebraic numbers giving algebraic results is called the exceptional set of that function.^[7]^[8] Formally it is defined by:

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

In many instances the exceptional set is fairly small. For example, ${\mathcal {E}}(\exp )=\{0\},$ this was proved by Lindemann in 1882. 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 if it can be calculated then it can often lead to results in transcendental number theory. Here are some other known exceptional sets:

Klein's j-invariant ${\mathcal {E}}(j)=\left\{\alpha \in {\mathcal {H}}\,:\,[\mathbb {Q} (\alpha ):\mathbb {Q} ]=2\right\},$ where ⁠ ${\mathcal {H}}$ ⁠ is the upper half-plane, and ⁠ $[\mathbb {Q} (\alpha ):\mathbb {Q} ]$ ⁠ is the degree of the number field ⁠ $\mathbb {Q} (\alpha ).$ ⁠ This result is due to Theodor Schneider.^[9]
Exponential function in base 2: ${\mathcal {E}}(2^{x})=\mathbb {Q} ,$ This result is a corollary of the Gelfond–Schneider theorem, which states that if $\alpha \neq 0,1$ is algebraic, and $\beta$ is algebraic and irrational then $\alpha ^{\beta }$ is transcendental. Thus the function $2 x$ could be replaced by $c x$ for any algebraic $c$ not equal to 0 or 1. Indeed, we have: ${\mathcal {E}}(x^{x})={\mathcal {E}}\left(x^{\frac {1}{x}}\right)=\mathbb {Q} \setminus \{0\}.$
A consequence of Schanuel's conjecture in transcendental number theory would be that ${\mathcal {E}}\left(e^{e^{x}}\right)=\emptyset .$
A function with empty exceptional set that does not require assuming Schanuel's conjecture is $f(x)=\exp(1+\pi x).$

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 $A$ .^[10] 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.^[11]

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, $log(5 metres)$ is a nonsensical expression, unlike $log(5 metres / 3 metres)$ or $log(3) metres$ . 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.

Related Research Articles

An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer coefficients. For example, the golden ratio, $, is an algebraic number, because it is a root of the polynomial x 2 - x - 1 . That is, it is a value for x for which the polynomial evaluates to zero. As another example, the complex number is algebraic because it is a root of x 4 + 4 .$

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted $i$ , called the imaginary unit and satisfying the equation $; every complex number can be expressed in the form, where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number, a is called the real part, and b is called the imaginary part . The set of complex numbers is denoted by either of the symbols or C . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.$

The number $e$ is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted $. Alternatively, e can be called Napier's constant after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.$

<span class="mw-page-title-main">Exponential function</span> Mathematical function, denoted exp(x) or e^x

The exponential function is a mathematical function denoted by $or . Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the operation of taking powers of a number, but various modern definitions allow it to be rigorously extended to all real arguments, including irrational numbers. Its ubiquitous occurrence in pure and applied mathematics led mathematician Walter Rudin to consider the exponential function to be "the most important function in mathematics".$

In mathematics, a transcendental number is a real or complex number that is not algebraic – that is, not the root of a non-zero polynomial with integer coefficients. The best-known transcendental numbers are $π$ and $e$ . The quality of a number being transcendental is called transcendence.

In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

<span class="mw-page-title-main">Exponentiation</span> Arithmetic operation

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as $b n$ , where $b$ is the base and $n$ is the power; this is pronounced as " $b$ (raised) to the $n$ ". When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b n$ is the product of multiplying $n$ bases:

In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following:

<span class="mw-page-title-main">Tetration</span> Arithmetic operation

In mathematics, tetration is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though Knuth's up arrow notation $and the left-exponent x b are common.$

In mathematics, an expression or equation is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.

Transcendental number theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.

In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.

In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.

In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.

In applied mathematics, a transcendental equation is an equation over the real numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. Examples include:

In mathematics, specifically algebraic geometry, a period or algebraic period is a complex number that can be expressed as an integral of an algebraic function over an algebraic domain. The periods are a class of numbers which includes, alongside the algebraic numbers, many well known mathematical constants such as the number π. Sums and products of periods remain periods, such that the periods $form a ring.$

In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions that appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value zero for many arguments, or having a zero of high order at some point.

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In transcendental number theory, a mathematical discipline, Baker's theorem gives a lower bound for the absolute value of linear combinations of logarithms of algebraic numbers. Nearly fifteen years earlier, Alexander Gelfond had considered the problem with only integer coefficients to be of "extraordinarily great significance". The result, proved by Alan Baker, subsumed many earlier results in transcendental number theory. Baker used this to prove the transcendence of many numbers, to derive effective bounds for the solutions of some Diophantine equations, and to solve the class number problem of finding all imaginary quadratic fields with class number 1.

References

↑ Townsend, E.J. (1915). Functions of a Complex Variable. H. Holt. p. 300. OCLC 608083625.
↑ Hazewinkel, Michiel (1993). Encyclopedia of Mathematics. Vol. 9. pp. 236.
↑ Waldschmidt, M. (2000). Diophantine approximation on linear algebraic groups. Springer. ISBN 978-3-662-11569-5.
↑ Pedersen, Olaf (1974). Survey of the Almagest. Odense University Press. p. 84. ISBN 87-7492-087-1.
↑ Rubel, Lee A. (November 1989). "A Survey of Transcendentally Transcendental Functions". The American Mathematical Monthly. 96 (9): 777–788. doi:10.1080/00029890.1989.11972282. JSTOR 2324840.
↑ van der Poorten, A.J. (1968). "Transcendental entire functions mapping every algebraic number field into itself". J. Austral. Math. Soc. 8 (2): 192–8. doi: 10.1017/S144678870000522X . S2CID 121788380.
↑ Marques, D.; Lima, F.M.S. (2010). "Some transcendental functions that yield transcendental values for every algebraic entry". arXiv: 1004.1668v1 [math.NT].
↑ Archinard, N. (2003). "Exceptional sets of hypergeometric series". Journal of Number Theory. 101 (2): 244–269. doi:10.1016/S0022-314X(03)00042-8.
↑ Schneider, T. (1937). "Arithmetische Untersuchungen elliptischer Integrale". Math. Annalen. 113: 1–13. doi:10.1007/BF01571618. S2CID 121073687.
↑ Waldschmidt, M. (2009). "Auxiliary functions in transcendental number theory". The Ramanujan Journal. 20 (3): 341–373. arXiv: 0908.4024 . doi:10.1007/s11139-009-9204-y. S2CID 122797406.
↑ Wilkie, A.J. (1998). "An algebraically conservative, transcendental function". Paris VII Preprints. 66.

External links

Definition of "Transcendental function" in the Encyclopedia of Math

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[1] Townsend, E.J. (1915). Functions of a Complex Variable. H. Holt. p. 300. OCLC 608083625.

[2] Hazewinkel, Michiel (1993). Encyclopedia of Mathematics. Vol. 9. pp. 236.

[3] Waldschmidt, M. (2000). Diophantine approximation on linear algebraic groups. Springer. ISBN 978-3-662-11569-5.

[4] Pedersen, Olaf (1974). Survey of the Almagest. Odense University Press. p. 84. ISBN 87-7492-087-1.

[5] Rubel, Lee A. (November 1989). "A Survey of Transcendentally Transcendental Functions". The American Mathematical Monthly. 96 (9): 777–788. doi:10.1080/00029890.1989.11972282. JSTOR 2324840.

[6] van der Poorten, A.J. (1968). "Transcendental entire functions mapping every algebraic number field into itself". J. Austral. Math. Soc. 8 (2): 192–8. doi: 10.1017/S144678870000522X . S2CID 121788380.

[7] Marques, D.; Lima, F.M.S. (2010). "Some transcendental functions that yield transcendental values for every algebraic entry". arXiv: 1004.1668v1 [math.NT].

[8] Archinard, N. (2003). "Exceptional sets of hypergeometric series". Journal of Number Theory. 101 (2): 244–269. doi:10.1016/S0022-314X(03)00042-8.

[9] Schneider, T. (1937). "Arithmetische Untersuchungen elliptischer Integrale". Math. Annalen. 113: 1–13. doi:10.1007/BF01571618. S2CID 121073687.

[10] Waldschmidt, M. (2009). "Auxiliary functions in transcendental number theory". The Ramanujan Journal. 20 (3): 341–373. arXiv: 0908.4024 . doi:10.1007/s11139-009-9204-y. S2CID 122797406.

[11] Wilkie, A.J. (1998). "An algebraically conservative, transcendental function". Paris VII Preprints. 66.

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