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| A '''transcendental function''' is a [[function (mathematics)|function]] that does not satisfy a [[polynomial]] equation whose [[coefficient]]s are themselves polynomials, in contrast to an [[algebraic function]], which does satisfy such an equation.<ref>E. J. Townsend, ''Functions of a Complex Variable'', BiblioLife, LLC, (2009).</ref> (The polynomials are sometimes required to have [[rational number|rational]] coefficients.) In other words, a '''transcendental function''' is a function that "[[wiktionary:transcend|transcends]]" [[algebra]] in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.
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| Examples of transcendental functions include the [[exponential function]], the [[logarithm]], and the [[trigonometric function]]s.
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| ==Definition==
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| Formally, an [[analytic function]] ƒ(''z'') of one real or complex variable ''z'' is transcendental if it is [[algebraic independence|algebraically independent]] of that variable.<ref>M. Waldschmidt, ''Diophantine approximation on linear algebraic groups'', Springer (2000).</ref> This can be extended to functions of several variables.
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| ==Examples==
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| The following functions are transcendental: | |
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| :<math>f_1(x) = x^\pi \ </math>
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| :<math>f_2(x) = c^x, \ c \ne 0, 1</math>
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| :<math>f_3(x) = x^{x} = {{^2}x} \ </math>
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| :<math>f_4(x) = x^{\frac{1}{x}} \ </math>
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| :<math>f_5(x) = \log_c x, \ c \ne 0, 1</math>
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| :<math>f_6(x) = \sin{x}</math>
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| Note that in particular for ƒ<sub>2</sub> if we set c equal to ''e'', the [[exponential function|base of the natural logarithm]], then we get that ''e<sup>x</sup>'' is a transcendental function. Similarly, if we set ''c'' equal to ''e'' in ƒ<sub>5</sub>, then we get that ln(''x''), the [[natural logarithm]], is a transcendental function. For more information on the second notation of ƒ<sub>3</sub>, see [[tetration]].
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| ==Algebraic and transcendental functions==
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| {{details|elementary function (differential algebra)}}
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| The [[logarithm]] and the [[exponential function]] are examples of transcendental functions. ''Transcendental function'' is a term often used to describe the [[trigonometric function]]s ([[sine]], [[cosine]], [[tangent (trigonometric function)|tangent]], their reciprocals [[Trigonometric functions#Reciprocal functions|cotangent, secant, and cosecant]], the now little-used [[versine|versine, haversine, and coversine]], their analogs the [[hyperbolic functions]] and so forth). | |
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| A function that is not transcendental is said to be '''algebraic'''. Examples of algebraic functions are [[rational functions]] and the [[square root]] function.
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| 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 [[Multiplicative inverse|reciprocal function]] in an effort to find the area of a [[hyperbolic sector]]. Thus the [[hyperbolic angle]] and the [[hyperbolic function]]s sinh, cosh, and tanh are all transcendental.
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| [[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.
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| ==Dimensional analysis==
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| 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 [[logarithm]]ic identity to get log(10) + log(m), which highlights the problem: applying a non-algebraic operation to a dimension creates meaningless results.
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| ==Exceptional set==
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| 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 function|entire transcendental function]]s 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'') = ''e<sup>z</sup>'', 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,<ref>D. Marques, F. M. S. Lima, ''Some transcendental functions that yield transcendental values for every algebraic entry'', (2010) {{arxiv|1004.1668v1}}.</ref><ref>N. Archinard, ''Exceptional sets of hypergeometric series'', Journal of Number Theory '''101''' Issue 2 (2003), pp.244–269.</ref> that is the set
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| :<math>\mathcal{E}(f)=\{\alpha\in\overline{\mathbf{Q}}\,:\,f(\alpha)\in\overline{\mathbf{Q}}\}.</math> | |
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| If this set can be calculated then it can often lead to results in [[transcendence theory]]. For example, [[Ferdinand von Lindemann|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(''i''π) = -1 is algebraic we know that ''i''π cannot be algebraic. Since ''i'' is algebraic this implies that ''π'' is a [[transcendental number]].
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| In general, finding the exceptional set of a function is a difficult problem, but it has been calculated for some functions:
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| *<math>\mathcal{E}(\exp)=\{0\}</math>,
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| *<math>\mathcal{E}(j)=\{\alpha\in\mathbf{H}\,:\,[\mathbf{Q}(\alpha): \mathbf{Q}]=2\}</math>,
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| **Here ''j'' is Klein's [[j-invariant|''j''-invariant]], '''H''' is the [[upper half-plane]], and ['''Q'''(α): '''Q'''] is the [[Degree of a field extension|degree]] of the [[Algebraic number field|number field]] '''Q'''(α). This result is due to [[Theodor Schneider]].<ref>T. Schneider, ''Arithmetische Untersuchungen elliptischer Integrale'', Math. Annalen '''113''' (1937), pp.1–13.</ref>
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| *<math>\mathcal{E}(2^{x})=\mathbf{Q}</math>,
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| **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 α<sup>β</sup> is transcendental. Thus the function 2<sup>''x''</sup> could be replaced by ''c<sup>x</sup>'' for any algebraic ''c'' not equal to 0 or 1. Indeed, we have:
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| *<math>\mathcal{E}(x^x)=\mathcal{E}(x^{\frac{1}{x}})=\mathbf{Q}\setminus\{0\}.</math>
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| *A consequence of [[Schanuel's conjecture]] in [[transcendence theory|transcendental number theory]] would be that <math>\mathcal{E}(e^{e^x})=\emptyset.</math>
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| *A function with empty exceptional set that doesn't require one to assume this conjecture is the function ƒ(''x'') = exp(1 + π''x'').
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| 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''.<ref>M. Waldschmidt, ''Auxiliary functions in transcendental number theory'', The Ramanujan journal '''20''' no3, (2009), pp.341–373.</ref> 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: '''R''' → '''R''' that is analytic everywhere but whose transcendence cannot be detected by any [[First-order logic|first-order]] method.<ref>A. Wilkie, ''An algebraically conservative, transcendental function'', Paris VII preprints, number 66, 1998.</ref>
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| ==See also==
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| *[[Algebraic function]]
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| *[[Analytic function]]
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| *[[Complex function]]
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| *[[Function (mathematics)]]
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| *[[Generalized function]]
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| *[[List of special functions and eponyms]]
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| *[[List of types of functions]]
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| *[[Polynomial]]
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| *[[Rational function]]
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| *[[Special functions]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.encyclopediaofmath.org/index.php/Transcendental_function Definition of "Transcendental function" in the Encyclopedia of Math]
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| [[Category:Analytic functions]]
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| [[Category:Functions and mappings]]
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| [[Category:Meromorphic functions]]
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| [[Category:Special functions]]
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| [[Category:Types of functions]]
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During the fall of 2011, Concert tours: Luke Bryan And that had a remarkable selection of , which include Urban. “It’s much like you’re getting a authorization to look to another level, says all those artists that had been a part of the Concert toursabove in a greater degree of designers.” It covered as the most successful tours in its ten-year record.
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