Editing Transcendental number (section)

==Numbers proven to be transcendental==
Numbers proven to be transcendental:

* [[pi|{{mvar|π}}]] (by the [[Lindemann–Weierstrass theorem]]).
* {{math|<math>e^a</math>}} if {{Math|<math>a</math>}} is [[Algebraic number|algebraic]] and nonzero (by the Lindemann–Weierstrass theorem), in particular [[E (mathematical constant)|Euler's number]] {{mvar|e}}.
* {{math|<math> e^{\pi \sqrt n} </math>}} where {{math|<math>n</math>}} is a positive integer; in particular [[Gelfond's constant]] {{math|<math>e^\pi</math>}} (by the [[Gelfond–Schneider theorem]]).
* Algebraic combinations of {{math|<math>\pi </math>}} and {{math|<math> e^{\pi \sqrt n} , n\in\mathbb Z^{+}</math>}} such as {{math|<math> \pi + e^{\pi}</math>}} and {{math|<math> \pi  e^{\pi}</math>}} (following from their [[algebraic independence]]).<ref name=":2">{{Cite journal |last=Nesterenko |first=Yu V |date=1996-10-31 |title=Modular functions and transcendence questions |url=https://iopscience.iop.org/article/10.1070/SM1996v187n09ABEH000158 |journal=Sbornik: Mathematics |volume=187 |issue=9 |pages=1319–1348 |doi=10.1070/SM1996v187n09ABEH000158 |bibcode=1996SbMat.187.1319N |issn=1064-5616}}</ref>
* {{math|<math>a^b</math>}} where {{Math|<math>a</math>}} is algebraic but not 0 or 1, and {{Math|<math>b</math>}} is irrational algebraic, in particular the [[Gelfond–Schneider constant]] <math>2^{\sqrt{2}}</math> (by the Gelfond–Schneider theorem).
* The [[natural logarithm]] {{math|<math>\ln(a)</math>}} if {{math|<math>a</math>}} is algebraic and not equal to 0 or 1, for any branch of the logarithm function (by the Lindemann–Weierstrass theorem).
* {{math|<math>\log_b(a)</math>}} if {{math|<math>a</math>}} and {{math|<math>b</math>}} are positive integers not both powers of the same integer, and {{math|<math>a</math>}} is not equal to 1 (by the Gelfond–Schneider theorem).
* All numbers of the form <math>\pi + \beta_1 \ln (a_1) + \cdots + \beta_n \ln (a_n)</math> are transcendental, where <math>\beta_j</math> are algebraic for all <math>1 \leq j \leq n</math> and <math>a_j</math> are non-zero algebraic for all <math>1 \leq j \leq n</math> (by [[Baker's theorem]]).

*The [[trigonometric functions]] {{math|<math>\sin(x), \cos(x), ...</math>}} and their [[Hyperbolic functions|hyperbolic counterparts]], for any nonzero algebraic number {{math|<math>x</math>}}, expressed in [[radian]]s (by the Lindemann–Weierstrass theorem).
*Non-zero results of the [[inverse trigonometric functions]] {{math|<math>\arcsin(x), \arccos(x), ...</math>}} and their [[Inverse hyperbolic functions|hyperbolic counterparts]], for any algebraic number {{math|<math>x</math>}} (by the Lindemann–Weierstrass theorem).
*<math>\pi^{-1}{\arctan(x)}</math>, for rational {{math|<math>x</math>}} such that <math>x \notin \{0,\pm{1}\}</math>.<ref name=":1">{{Cite web |last=Weisstein |first=Eric W. |title=Transcendental Number |url=https://mathworld.wolfram.com/TranscendentalNumber.html |access-date=2023-08-09 |website=mathworld.wolfram.com |language=en}}</ref>
*The [[Fixed-point iteration#Attracting fixed points|fixed point]] of the cosine function (also referred to as the [[Dottie number]] {{math|<math>d</math>}}) – the unique real solution to the equation {{math|<math>\cos(x)=x</math>}}, where {{math|<math>x</math>}} is in radians (by the Lindemann–Weierstrass theorem).<ref name="wolfram_dottie">{{cite web|last1=Weisstein|first1=Eric W.|title=Dottie Number|url=http://mathworld.wolfram.com/DottieNumber.html|website=Wolfram MathWorld|publisher=Wolfram Research, Inc.|access-date=23 July 2016}}</ref>
*{{math|<math>W(a)</math>}} if {{math|<math>a</math>}} is algebraic and nonzero, for any branch of the [[Lambert W function|Lambert W Function]] (by the Lindemann–Weierstrass theorem), in particular the [[omega constant]] {{math|Ω}}.
* {{math|<math>W(r,a)</math>}} if both {{math|<math>a</math>}} and the order {{math|<math>r</math>}} are algebraic such that <math>a \neq 0</math>, for any branch of the generalized Lambert W function.<ref>{{Cite arXiv |eprint=1408.3999 |class=math.CA |first1=István |last1=Mező |first2=Árpád |last2=Baricz |title=On the generalization of the Lambert W function |date=June 22, 2015}}</ref>
* {{math|<math>\sqrt x _s</math>}}, the [[Tetration#Square super-root|square super-root]] of any natural number is either an integer or transcendental (by the Gelfond–Schneider theorem).
* Values of the [[gamma function]] of rational numbers that are of the form <math>\Gamma(n/2),\Gamma(n/3),\Gamma(n/4)</math> or <math>\Gamma(n/6)</math>.<ref>{{Cite book |last=Chudnovsky |first=G. |title=Contributions to the theory of transcendental numbers |date=1984 |publisher=American Mathematical Society |isbn=978-0-8218-1500-7 |series=Mathematical surveys and monographs |location=Providence, R.I |language=en, ru}}</ref>
* Algebraic combinations of {{math|<math>\pi </math>}} and {{math|<math>\Gamma(1/3)</math>}} or of {{math|<math>\pi </math>}} and {{math|<math>\Gamma(1/4)</math>}} such as the [[lemniscate constant]] <math>\varpi</math> (following from their respective algebraic independences).<ref name=":2" />
* The values of [[Beta function]] <math>\Beta(a,b)</math> if <math>a, b</math> and <math>a+b</math> are non-integer rational numbers.<ref name=":3">{{Cite web |last=Waldschmidt |first=Michel |date=September 7, 2005 |title=Transcendence of Periods: The State of the Art |url=https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TranscendencePeriods.pdf |website=webusers.imj-prg.fr}}</ref>
* The [[Bessel function|Bessel function of the first kind]] {{math|<math>J_\nu(x)</math>}}, its first derivative, and the quotient <math>\tfrac{J'_\nu (x)}{J_\nu (x)}</math> are transcendental when ''{{math|<math>\nu</math>}}'' is rational and ''{{math|<math>x</math>}}'' is algebraic and nonzero,<ref>{{cite book |last1=Siegel |first1=Carl L. |title=On Some Applications of Diophantine Approximations |chapter=Über einige Anwendungen diophantischer Approximationen: Abhandlungen der Preußischen Akademie der Wissenschaften. Physikalisch-mathematische Klasse 1929, Nr. 1 |date=2014 |publisher=Scuola Normale Superiore |isbn=978-88-7642-520-2 |pages=81–138 |chapter-url=https://doi.org/10.1007/978-88-7642-520-2_2 |language=de |doi=10.1007/978-88-7642-520-2_2 }}</ref> and all nonzero roots of {{math|<math>J_\nu(x)</math>}} and {{math|<math>J'_\nu(x)</math>}} are transcendental when ''{{math|<math>\nu</math>}}'' is rational.<ref>{{cite journal |last1=Lorch |first1=Lee |last2=Muldoon |first2=Martin E. |title=Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |journal=International Journal of Mathematics and Mathematical Sciences |date=1995 |volume=18 |issue=3 |pages=551–560 |doi=10.1155/S0161171295000706 |doi-access=free }}</ref>
* The number <math>\tfrac{\pi}{2} \tfrac{Y_0 (2)}{J_0 (2)} - \gamma</math>, where {{math|<math>Y_\alpha(x)</math>}} and {{math|<math>J_\alpha(x)</math>}} are Bessel functions and {{math|<math>\gamma</math>}} is the [[Euler–Mascheroni constant]].<ref>{{Cite journal |last1=Mahler |first1=Kurt |last2=Mordell |first2=Louis Joel |date=1968-06-04 |title=Applications of a theorem by A. B. Shidlovski |url=https://royalsocietypublishing.org/doi/10.1098/rspa.1968.0111 |journal=Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences |volume=305 |issue=1481 |pages=149–173 |bibcode=1968RSPSA.305..149M |doi=10.1098/rspa.1968.0111 |s2cid=123486171}}</ref><ref>{{Cite journal |last=Lagarias |first=Jeffrey C. |date=2013-07-19 |title=Euler's constant: Euler's work and modern developments |journal=Bulletin of the American Mathematical Society |volume=50 |issue=4 |pages=527–628 |arxiv=1303.1856 |doi=10.1090/S0273-0979-2013-01423-X |issn=0273-0979 |doi-access=free}}</ref>
* Values of the [[Reciprocal Fibonacci constant|Fibonacci zeta function]] at the positive even argument.<ref name="Murty2013">{{citation 
|last = Murty | first = M. Ram
 | editor1-last = Prasad | editor1-first = D.
 | editor2-last = Rajan | editor2-first = C. S.
 | editor3-last = Sankaranarayanan | editor3-first = A.
 | editor4-last = Sengupta | editor4-first = J.
 | contribution = The Fibonacci zeta function | isbn = 978-93-80250-49-6 | mr = 3156859 | pages = 409–425
 | publisher = Tata Institute of Fundamental Research
 | series = Tata Institute of Fundamental Research Studies in Mathematics
 | title = Automorphic representations and {{mvar|L}}-functions
 | volume = 22 | year = 2013}}</ref>
* Any [[Liouville number]], in particular: Liouville's constant <math>\sum_{k=1}^\infty\frac1{10^{k!}}</math>.
* Numbers with [[irrationality measure]] larger than 2, such as the [[Champernowne constant]] <math>C_{10}</math> (by [[Roth's theorem]]).
* Numbers artificially constructed not to be [[Period (algebraic geometry)|algebraic periods]].<ref>{{cite arXiv |eprint=0805.0349 |class=math.AG |first=Masahiko |last=Yoshinaga |title=Periods and elementary real numbers |date=2008-05-03}}</ref>
* Any [[non-computable number]], in particular: [[Chaitin's constant]].
* Constructed irrational numbers which are not simply [[Normal number|normal]] in any base.{{sfn|Bugeaud|2012|page=113}}
* Any number for which the digits with respect to some fixed base form a [[Sturmian word]].<ref>{{harvnb|Pytheas Fogg|2002}}</ref>
* The [[Prouhet–Thue–Morse constant]]<ref>{{harvnb|Mahler|1929}}; {{harvnb|Allouche|Shallit|2003|p=387}}</ref> and the related rabbit constant.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Rabbit Constant |url=https://mathworld.wolfram.com/ |access-date=2023-08-09 |website=mathworld.wolfram.com |language=en}}</ref>
* The [[Komornik–Loreti constant]].<ref>{{citation |last1=Allouche |first1=Jean-Paul |title=The Komornik–Loreti constant is transcendental |journal=American Mathematical Monthly |volume=107 |issue=5 |pages=448–449 |year=2000 |doi=10.2307/2695302 |jstor=2695302 |mr=1763399 |last2=Cosnard |first2=Michel}}</ref>
* The [[Regular paperfolding sequence|paperfolding constant]] (also named as "Gaussian Liouville number").<ref>{{Cite web |title=A143347 - OEIS |url=https://oeis.org/A143347 |access-date=2023-08-09 |website=oeis.org}}</ref>
* The values of the infinite series with fast [[Rate of convergence|convergence rate]] as defined by Y. Gao and J. Gao, such as <math>\sum_{n=1}^\infty \frac{3^n}{2^{3^n}}</math>.<ref>{{Cite web |title=A140654 - OEIS |url=https://oeis.org/A140654 |access-date=2023-08-12 |website=oeis.org}}</ref>
* Any number of the form <math>\sum_{n=0}^\infty \frac{E_n(\beta^{r^n})}{F_n(\beta^{r^n})}</math> (where <math>E_n(z)</math>, <math>F_n(z)</math> are polynomials in variables <math>n</math> and <math>z</math>, <math>\beta</math> is algebraic and <math>\beta \neq 0</math>, <math>r</math> is any integer greater than 1).<ref>{{Cite journal |last=Kurosawa |first=Takeshi |date=2007-03-01 |title=Transcendence of certain series involving binary linear recurrences |journal=Journal of Number Theory |language=en |volume=123 |issue=1 |pages=35–58 |doi=10.1016/j.jnt.2006.05.019 |issn=0022-314X |doi-access=free}}</ref>
* Numbers of the form <math>\sum_{k=0}^\infty 10^{-b^k}</math> and <math>\sum_{k=0}^\infty 10^{-\left\lfloor b^{k} \right\rfloor}</math> For {{math|b > 1}} where <math>b \mapsto\lfloor b \rfloor</math> is the [[floor function]].<ref name="Kempner" /><ref>{{Cite arXiv |eprint=1303.1685 |class=math.NT |first=Boris |last=Adamczewski |title=The Many Faces of the Kempner Number |date=March 2013}}</ref><ref name="Sha1999">{{harvnb|Shallit|1996}}</ref><ref>{{Cite journal |last1=Adamczewski |first1=Boris |last2=Rivoal |first2=Tanguy | authorlink2=Tanguy Rivoal |date=2009 |title=Irrationality measures for some automatic real numbers |url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/irrationality-measures-for-some-automatic-real-numbers/F89F4B7BBC9A06B6E9934FB2C3AFFE4D |journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]] |language=en |volume=147 |issue=3 |pages=659–678 |doi=10.1017/S0305004109002643 |bibcode=2009MPCPS.147..659A |issn=1469-8064}}</ref><ref name="Lox1988">{{harvnb|Loxton|1988}}</ref><ref>{{harvnb|Allouche|Shallit|2003|pp=385,403}}</ref>
* The numbers <math>\alpha =  3.3003300000...</math> and <math> \alpha^{-1} = 0.3030000030...</math> with only two different decimal digits whose nonzero digit positions are given by the [[Moser–de Bruijn sequence]] and its double.<ref>{{harvnb|Blanchard|Mendès France|1982}}</ref>
* The values of the [[Rogers–Ramanujan continued fraction|Rogers-Ramanujan continued fraction]] <math>R(q)</math> where <math>{{q}} \in \mathbb C</math> is algebraic and <math>0 < |q| < 1</math>.<ref>{{Cite journal |last1=Duverney |first1=Daniel |last2=Nishioka |first2=Keiji |last3=Nishioka |first3=Kumiko |last4=Shiokawa |first4=Iekata |date=1997 |title=Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers |journal=Proceedings of the Japan Academy, Series A, Mathematical Sciences |volume=73 |issue=7 |pages=140–142 |doi=10.3792/pjaa.73.140 |issn=0386-2194 |doi-access=free}}</ref> The lemniscatic values of [[theta function]] <math>\sum_{n=-\infty}^\infty q^{n^2}</math> (under the same conditions for <math>{{q}}</math>) are also transcendental.<ref>{{Cite journal |last=Bertrand |first=Daniel |date=1997 |title=Theta functions and transcendence |url=http://link.springer.com/10.1023/A:1009749608672 |journal=The Ramanujan Journal |volume=1 |issue=4 |pages=339–350 |doi=10.1023/A:1009749608672 |s2cid=118628723}}</ref>
* {{math|''[[j-invariant|j]]''(''q'')}} where <math>{{q}} \in \mathbb C</math> is algebraic but not imaginary quadratic (i.e, the [[Transcendental function|exceptional set]] of this function is the number field whose degree of [[Field extension|extension]] over <math>\mathbb Q</math> is 2).
* The constants <math>\epsilon_k</math> and <math>\nu_k</math> in the formula for first index of occurrence of [[Gijswijt's sequence]], where k is any integer greater than 1.<ref>{{cite arXiv |eprint=2209.04657 |first=Levi |last=van de Pol |title=The first occurrence of a number in Gijswijt's sequence|date=2022 |class=math.CO }}</ref>