(** 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 is_derive_2_Bessel1 (n : nat) (x : R) :
is_derive_n (Bessel1 n) 2 x
(((x/2)^(S (S n)) * PSeries (PS_derive (PS_derive (Bessel1_seq n))) ((x / 2) ^ 2))
+ ((INR (2*n+1)/2) * (x/2)^n * PSeries (PS_derive (Bessel1_seq n)) ((x / 2) ^ 2))
+ (INR (n * pred n) / 4 * (x / 2) ^ pred (pred n) * PSeries (Bessel1_seq n) ((x / 2) ^ 2))).