Editing Bessel function (section)

== Zeros of the Bessel function ==

=== Bourget's hypothesis ===
Bessel himself originally proved that for nonnegative integers {{mvar|n}}, the equation {{math|1=''J''<sub>''n''</sub>(''x'') = 0}} has an infinite number of solutions in {{mvar|x}}.<ref>Bessel, F. (1824), article 14.</ref> When the functions {{math|''J''<sub>''n''</sub>(''x'')}} are plotted on the same graph, though, none of the zeros seem to coincide for different values of {{mvar|n}} except for the zero at {{math|1=''x'' = 0}}. This phenomenon is known as '''Bourget's hypothesis''' after the 19th-century French mathematician who studied Bessel functions. Specifically it states that for any integers {{math|''n'' ≥ 0}} and {{math|''m'' ≥ 1}}, the functions {{math|''J<sub>n</sub>''(''x'')}} and {{math|''J''<sub>''n'' + ''m''</sub>(''x'')}} have no common zeros other than the one at {{math|1=''x'' = 0}}. The hypothesis was proved by [[Carl Ludwig Siegel]] in 1929.<ref>Watson, pp. 484–485.</ref>

=== Transcendence ===
Siegel proved in 1929 that when ''ν'' is rational, all nonzero roots of {{math|''J''<sub>''ν''</sub>(x)}} and {{math|''J''{{'}}<sub>''ν''</sub>(x)}} are [[transcendental number|transcendental]],<ref name="lorch"/> as are all the roots of {{math|''K''<sub>''ν''</sub>(x)}}.<ref name="euclid"/> It is also known that all roots of the higher derivatives <math>J_\nu^{(n)}(x)</math> for {{math|''n'' ≤ 18}} are transcendental, except for the special values <math>J_1^{(3)}(\pm\sqrt3) = 0</math> and <math>J_0^{(4)}(\pm\sqrt3) = 0</math>.<ref name="lorch">{{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>

=== Numerical approaches ===
For numerical studies about the zeros of the Bessel function, see {{harvtxt|Gil|Segura|Temme|2007}}, {{harvtxt|Kravanja|Ragos|Vrahatis|Zafiropoulos|1998}} and {{harvtxt|Moler|2004}}.

=== Numerical values ===
The first zeros in J<sub>0</sub> (i.e., j<sub>0,1</sub>, j<sub>0,2</sub> and j<sub>0,3</sub>) occur at arguments of approximately 2.40483, 5.52008 and 8.65373, respectively.<ref>Abramowitz & Stegun, p409</ref>