Editing Transcendental function

{{Short description|Analytic function that does not satisfy a polynomial equation}}

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]].<ref>{{cite book |first=E.J. |last=Townsend |title=Functions of a Complex Variable |publisher=H. Holt |location= |date=1915 |oclc=608083625 |pages=300 |url=}}</ref><ref>{{cite book |first=Michiel |last=Hazewinkel |title=Encyclopedia of Mathematics |publisher= |volume=9 |date=1993 |isbn= |pages=[https://books.google.com/books?id=1ttmCRCerVUC&pg=PA236 236] |url=}}</ref> 

Examples of transcendental functions include the [[exponential function]], the [[logarithm]] function, the [[hyperbolic function]]s, and the [[trigonometric function]]s. Equations over these expressions are called [[transcendental equation]]s.

==Definition==
Formally, an [[analytic function]] <math>f</math> of one [[real number|real]] or [[complex number|complex]] variable is '''transcendental''' if it is [[algebraically independent]] of that variable.<ref>{{cite book |first=M. |last=Waldschmidt |title=Diophantine approximation on linear algebraic groups |publisher=Springer |location= |date=2000 |isbn=978-3-662-11569-5 |pages= |url={{GBurl|Wrj0CAAAQBAJ|pg=PR9}}}}</ref> This means the function does not satisfy any polynomial equation. For example, the function <math>f</math> given by

:<math>f(x)=\frac{ax+b}{cx+d}</math> for all <math>x</math>

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

:<math>(ax+b)-(cx+d)f(x)=0</math>.

Similarly, the function <math>f</math> that satisfies the equation

:<math>f(x)^5+f(x)=x</math> for all <math>x</math>

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 [[Function of several real variables|functions of several variables]].

==History==
The transcendental functions [[sine]] and [[cosine]] were [[trigonometric tables|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:

{{quote|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]].<ref>{{cite book |author-link=Olaf Pedersen |first=Olaf |last=Pedersen |title=Survey of the Almagest |publisher=[[Odense University Press]] |location= |date=1974 |isbn=87-7492-087-1 |pages=84 |url=}}</ref>}}

A revolutionary understanding of these [[circular function]]s 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 function]]s through [[quadrature (mathematics)|quadrature]] of the [[rectangular hyperbola]] {{math|1=''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 (exponentiation)|base]] is [[e (mathematical constant)|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 {{nowrap|<math> \exp (x) = e^x</math>.}} Euler identified it with the [[infinite series]] {{nowrap|<math display="inline">\sum_{k=0} ^{\infty} x^k / k ! </math>,}} where {{math|''k''!}} denotes the [[factorial]] of {{mvar|k}}.

The even and odd terms of this series provide sums denoting {{math|cosh(''x'')}} and {{math|sinh(''x'')}}, so that <math>e^x = \cosh x + \sinh x.</math> These transcendental [[hyperbolic function]]s can be converted into circular functions sine and cosine by introducing {{math|(−1)<sup>''k''</sup>}} 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:

<math display="block">\begin{align}
f_1(x) &= x^\pi \\[2pt]
f_2(x) &= e^x \\[2pt]
f_3(x) &= \log_e{x} \\[2pt]
f_4(x) &= \cosh{x} \\
f_5(x) &= \sinh{x} \\
f_6(x) &= \tanh{x} \\
f_7(x) &= \sinh^{-1}{x} \\[2pt]
f_8(x) &= \tanh^{-1}{x} \\[2pt]
f_9(x) &= \cos{x} \\
f_{10}(x) &= \sin{x} \\
f_{11}(x) &= \tan{x} \\
f_{12}(x) &= \sin^{-1}{x} \\[2pt]
f_{13}(x) &= \tan^{-1}{x} \\[2pt]
f_{14}(x) &= x! \\
f_{15}(x) &= 1/x! \\[2pt]
f_{16}(x) &= x^x \\[2pt]
\end{align}</math>

For the first function <math>f_1(x)</math>, the exponent ''<math>\pi</math>'' can be replaced by any other irrational number, and the function will remain transcendental. For the second and third functions <math>f_2(x)</math> and <math>f_3(x)</math>, the base ''<math>e</math>'' can be replaced by any other positive real number base not equaling 1, and the functions will remain transcendental. Functions 4-8 denote the hyperbolic trigonometric functions, while functions 9-13 denote the circular trigonometric functions. The fourteenth function <math>f_{14}(x)</math> denotes the analytic extension of the factorial function via the [[gamma function]], and <math>f_{15}(x)</math> is its reciprocal, an entire function. Finally, in the last function <math>f_{16}(x)</math>, the exponent <math>x</math> can be replaced by <math>kx</math> for any nonzero real <math>k</math>, and the function will remain transcendental.

==Algebraic and transcendental functions==
{{details|Elementary function (differential algebra)}}

The most familiar transcendental functions are the [[logarithm]], the [[exponential function|exponential]] (with any non-trivial base), the [[trigonometric function|trigonometric]], and the [[hyperbolic functions]], and the [[inverse function|inverses]] of all of these. Less familiar are the [[special functions]] of [[mathematical analysis|analysis]], such as the [[gamma function|gamma]], [[elliptic function|elliptic]], and [[zeta function]]s, all of which are transcendental. The [[generalized hypergeometric function|generalized hypergeometric]] and [[Bessel function|Bessel]] functions are transcendental in general, but algebraic for some special parameter values.

Transcendental functions cannot be defined using only the operations of addition, subtraction, multiplication, division, and <math>n</math>th roots (where <math>n</math> 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]] function, but in general, algebraic functions cannot be defined as finite formulas of the elementary functions, as shown by the example above with <math>f(x)^5+f(x)=x</math> (see [[Abel–Ruffini theorem]]).

The [[indefinite integral]] of many algebraic functions is transcendental. For example, the integral <math>\int_{t=1}^x\frac{1}{t}dt</math> turns out to equal the logarithm function <math>log_e(x)</math>. Similarly, the limit or the infinite sum of many algebraic function sequences is transcendental. For example, <math>\lim_{n\to \infty}(1+x/n)^n</math> converges to the exponential function <math>e^x</math>, and the infinite sum <math>\sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}</math> turns out to equal the hyperbolic cosine function <math>\cosh x</math>. 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 equation]]s. Those that are not, such as the [[gamma function|gamma]] and the [[zeta function|zeta]] functions, are called ''transcendentally transcendental'' or ''[[hypertranscendental function|hypertranscendental]]'' functions.<ref>{{Cite journal|first=Lee A.|last=Rubel|title=A Survey of Transcendentally Transcendental Functions|journal=The American Mathematical Monthly|volume=96|number=9|date=November 1989|pages=777–788|doi=10.1080/00029890.1989.11972282|jstor=2324840}}</ref>

==Exceptional set==
If {{mvar|f}} is an algebraic function and <math>\alpha</math> is an [[algebraic number]] then {{math|''f'' (''&alpha;'')}} is also an algebraic number. The converse is not true: there are [[entire function|entire transcendental function]]s {{mvar|f}} such that {{math|''f'' (''&alpha;'')}} is an algebraic number for any algebraic {{mvar|&alpha;}}.<ref>{{cite journal |first=A.J. |last=van der Poorten |title=Transcendental entire functions mapping every algebraic number field into itself |journal=J. Austral. Math. Soc. |volume=8 |issue=2 |pages=192–8 |date=1968 |doi=10.1017/S144678870000522X |s2cid=121788380 |url=|doi-access=free }}</ref> For a given transcendental function the set of algebraic numbers giving algebraic results is called the '''exceptional set''' of that function.<ref>{{cite arXiv |first1=D. |last1=Marques |first2=F.M.S. |last2=Lima |title=Some transcendental functions that yield transcendental values for every algebraic entry |date=2010 |class=math.NT |eprint=1004.1668v1 }}</ref><ref>{{cite journal |first=N. |last=Archinard |title=Exceptional sets of hypergeometric series |journal=Journal of Number Theory |volume=101 |issue=2 |pages=244–269 |date=2003 |doi=10.1016/S0022-314X(03)00042-8 |doi-access= }}</ref> Formally it is defined by:

<math display="block">\mathcal{E}(f)=\left \{\alpha\in\overline{\Q}\,:\,f(\alpha)\in\overline{\Q} \right \}.</math>

In many instances the exceptional set is fairly small. For example, <math>\mathcal{E}(\exp) = \{0\},</math> this was proved by [[Ferdinand von Lindemann|Lindemann]] in 1882. In particular {{math|1=exp(1) = ''e''}} is transcendental. Also, since {{math|1=exp(''iπ'') = −1}} is algebraic we know that {{mvar|iπ}} cannot be algebraic. Since {{mvar|i}} is algebraic this implies that {{mvar|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|''j''-invariant]] <math display="block">\mathcal{E}(j) = \left\{\alpha\in\mathcal{H}\,:\,[\Q(\alpha): \Q] = 2 \right\},</math> where {{tmath|\mathcal H}} is the [[upper half-plane]], and {{tmath|[\Q(\alpha): \Q]}} is the [[Degree of a field extension|degree]] of the [[Algebraic number field|number field]] {{tmath|\Q(\alpha).}} This result is due to [[Theodor Schneider]].<ref>{{cite journal |first=T. |last=Schneider |title=Arithmetische Untersuchungen elliptischer Integrale |journal=Math. Annalen |volume=113 |issue= |pages=1–13 |date=1937 |doi=10.1007/BF01571618 |s2cid=121073687 |url=}}</ref>
* Exponential function in base 2: <math display="block">\mathcal{E}(2^x)=\Q,</math>This result is a corollary of the [[Gelfond–Schneider theorem]], which states that if <math>\alpha \neq 0,1</math> is algebraic, and <math>\beta</math> is algebraic and irrational then <math>\alpha^\beta</math> is transcendental. Thus the function {{math|2<sup>''x''</sup>}} could be replaced by {{mvar|c<sup>x</sup>}} for any algebraic {{mvar|c}} not equal to 0 or 1. Indeed, we have: <math display="block">\mathcal{E}(x^x) = \mathcal{E}\left(x^{\frac{1}{x}}\right)=\Q\setminus\{0\}.</math>
* A consequence of [[Schanuel's conjecture]] in transcendental number theory would be that <math>\mathcal{E}\left(e^{e^x}\right)=\emptyset.</math>
* A function with empty exceptional set that does not require assuming Schanuel's conjecture is <math>f(x) = \exp(1 + \pi x).</math>

While calculating the exceptional set for a given function is not easy, it is known that given ''any'' subset of the algebraic numbers, say {{mvar|A}}, there is a transcendental function whose exceptional set is {{mvar|A}}.<ref>{{cite journal |first=M. |last=Waldschmidt |title=Auxiliary functions in transcendental number theory |journal=The Ramanujan Journal |volume=20 |issue=3 |pages=341–373 |date=2009 |doi=10.1007/s11139-009-9204-y |arxiv=0908.4024 |s2cid=122797406 |url=}}</ref> The subset does not need to be proper, meaning that {{mvar|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]].<ref>{{cite journal |first=A.J. |last=Wilkie |author-link=Alex Wilkie |title=An algebraically conservative, transcendental function |journal=Paris VII Preprints |id=66 |date=1998 }}<!-- Conference abstract? Only cite in Google scholar --></ref>

==Dimensional analysis==
In [[dimensional analysis]], transcendental functions are notable because they make sense only when their argument is [[Dimensionless quantity|dimensionless]] (possibly after algebraic reduction). Because of this, transcendental functions can be an easy-to-spot source of dimensional errors. For example, {{math|log(5 metres)}} is a nonsensical expression, unlike {{math|log(5 metres / 3 metres)}} or {{math|log(3) metres}}. One could attempt to apply a [[logarithm]]ic identity to get {{math|log(5) + log(metres)}}, which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.

==See also==
*[[Complex function]]
*[[Function (mathematics)]]
*[[Generalized function]]
*[[List of special functions and eponyms]]
*[[List of types of functions]]
*[[Rational function]]
*[[Special functions]]

==References==
{{reflist}}

==External links==
{{wikibooks|Associative Composition Algebra|Transcendental paradigm|Transcendental functions}}
*[https://www.encyclopediaofmath.org/index.php/Transcendental_function Definition of "Transcendental function" in the Encyclopedia of Math]

[[Category:Analytic functions]]
[[Category:Functions and mappings]]
[[Category:Meromorphic functions]]
[[Category:Special functions]]
[[Category:Types of functions]]