Editing Transcendental function (section)

==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>