Editing Bessel function (section)

==== Bessel's integrals ====
Another definition of the Bessel function, for integer values of {{mvar|n}}, is possible using an integral representation:<ref name=Temme>{{cite book |last=Temme |first=Nico M. |title=Special Functions: An introduction to the classical functions of mathematical physics |year=1996 |publisher=Wiley |location=New York |isbn=0471113131 |pages=228–231 |edition=2nd print}}</ref>
<math display="block">J_n(x) = \frac{1}{\pi} \int_0^\pi \cos (n \tau - x \sin \tau) \,d\tau = \frac{1}{\pi} \operatorname{Re}\left(\int_{0}^\pi e^{i(n \tau-x \sin \tau )} \,d\tau\right),</math>
which is also called Hansen-Bessel formula.<ref>{{MathWorld|id=Hansen-BesselFormula|title=Hansen-Bessel Formula}}</ref>

This was the approach that Bessel used,<ref>Bessel, F. (1824).  The relevant integral is an unnumbered equation between equations 28 and 29.  Note that Bessel's <math>I^h_k</math> would today be written <math>J_h(k)</math>.</ref> and from this definition he derived several properties of the function. The definition may be extended to non-integer orders by one of Schläfli's integrals, for {{math|Re(''x'') > 0}}:<ref name=Temme /><ref>Watson, [https://books.google.com/books?id=Mlk3FrNoEVoC&pg=PA176 p. 176]</ref><ref>{{cite web |url=http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |title=Properties of Hankel and Bessel Functions |access-date=2010-10-18 |url-status=dead |archive-url=https://web.archive.org/web/20100923194031/http://www.math.ohio-state.edu/~gerlach/math/BVtypset/node122.html |archive-date=2010-09-23}}</ref><ref>{{cite web |url=https://www.nbi.dk/~polesen/borel/node15.html |title=Integral representations of the Bessel function |website=www.nbi.dk |access-date=25 March 2018 |archive-date=3 October 2022 |archive-url=https://web.archive.org/web/20221003054117/https://www.nbi.dk/~polesen/borel/node15.html |url-status=dead }}</ref><ref>Arfken & Weber, exercise 11.1.17.</ref>
<math display="block">J_\alpha(x) = \frac{1}{\pi} \int_0^\pi \cos(\alpha\tau - x \sin\tau)\,d\tau - \frac{\sin(\alpha\pi)}{\pi} \int_0^\infty e^{-x \sinh t - \alpha t} \, dt. </math>