(** This file is part of the Coquelicot formalization of real analysis in Coq: http://coquelicot.saclay.inria.fr/  Copyright (C) 2011-2015 Sylvie Boldo #<br /># Copyright (C) 2011-2015 Catherine Lelay #<br /># Copyright (C) 2011-2015 Guillaume Melquiond  This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.  This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details. *)
(** This file is an example of how to use power series. It defines and gives properties of the Bessel functions. *)
Definition Bessel1_seq (n k : nat) :=
(-1)^(k)/(INR (fact (k)) * INR (fact (n + (k)))).
Definition Bessel1 (n : nat) (x : R) :=
(x/2)^n * PSeries (Bessel1_seq n) ((x/2)^2).

Lemma Bessel1_correct (n : nat) (x : R) :
x^2 * Derive_n (Bessel1 n) 2 x + x * Derive (Bessel1 n) x + (x^2 - (INR n)^2) * Bessel1 n x = 0.