(** * Natural logarithm *)

(* Specifications: *)
Lemma ln_pow x n : 0 < x -> ln (x^n) = INR n * ln x.
intro Hx ;
induction n as [ | n IH].
Lemma is_lim_ln_p : is_lim (fun y => ln y) p_infty p_infty.
Lemma is_lim_ln_0 :
filterlim ln (at_right 0) (Rbar_locally m_infty).
Lemma is_lim_div_ln_p : is_lim (fun y => ln y / y) p_infty 0.
Lemma is_lim_div_ln1p_0 : is_lim (fun y => ln (1+y) / y) 0 1.
(** * Unnormalized sinc *)
Lemma is_lim_sinc_0 : is_lim (fun x => sin x / x) 0 1.
(** * ArcTan *)
Lemma CV_radius_atan : CV_radius (fun n => (-1)^n / (INR (S (n + n)))) = 1.
Lemma atan_Reals (x : R) : Rabs x < 1
-> atan x = x * PSeries (fun n => (-1)^n / (INR (S (n + n)))) (x ^ 2).