Editing Bessel function (section)

=== Transcendence ===
In 1929, [[Carl Ludwig Siegel]] proved that {{math|''J''<sub>''ν''</sub>(''x'')}}, {{math|''J''{{'}}<sub>''ν''</sub>(''x'')}}, and the [[logarithmic derivative]] {{math|{{sfrac|''J''{{'}}<sub>''ν''</sub>(''x'')|''J''<sub>''ν''</sub>(''x'')}}}} are [[transcendental number]]s when ''ν'' is rational and ''x'' is algebraic and nonzero.<ref>{{cite book |last1=Siegel |first1=Carl L. |title=On Some Applications of Diophantine Approximations: a translation of Carl Ludwig Siegel's Über einige Anwendungen diophantischer Approximationen by Clemens Fuchs, with a commentary and the article Integral points on curves: Siegel's theorem after Siegel's proof by Clemens Fuchs and Umberto Zannier |date=2014 |publisher=Scuola Normale Superiore |isbn=978-88-7642-520-2 |pages=81–138 |chapter-url=https://link.springer.com/chapter/10.1007/978-88-7642-520-2_2 |language=de |chapter=Über einige Anwendungen diophantischer Approximationen |doi=10.1007/978-88-7642-520-2_2}}</ref> The same proof also implies that <math> \Gamma(v+1)(2/x)^v J_{v}(x) </math> is transcendental under the same assumptions.<ref name="euclid">{{cite journal |last1=James |first1=R. D. |title=Review: Carl Ludwig Siegel, Transcendental numbers |journal=Bulletin of the American Mathematical Society |date=November 1950 |volume=56 |issue=6 |pages=523–526 |doi=10.1090/S0002-9904-1950-09435-X |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-56/issue-6/Review-Carl-Ludwig-Siegel-Transcendental-numbers/bams/1183515049.full|doi-access=free }}</ref>