70: 1'A XXVIII NOUVELLES EXPRESSIONS D'ALGÈBRE FONCTIONNELLE SIMPLE DES FONCTIONS TRIGONOMÉTRIQUES: FONCTIONS PLANCHER ET PLAFOND

© "Tous droits réservés" - 2012 par Cédric Christian Bernard Gagneux né le 19/07/64.

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XXVIII) EXPRESSIONS DES FONCTIONS TRIGONOMETRIQUES EQUIVALENTES A LA FONCTION DE LA PARTIE ENTIERE INFERIEURE OU SUPERIEURE D'UN NOMBRE ENTIER NATUREL

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1.a) Définition:
La partie entière (par défaut) inférieure de (x+1)/n est notée conventionnellement par la notation anglo-saxonne ⌊(x+1)/n⌋, appelée en anglais « floor », plancher, et la partie entière supérieure de x/n, appelée en anglais « ceiling », plafond est notée ⌈x/n⌉.

1.b) Expressions:

⌊n/x⌋=(n-(x/2+x*arctan(tan(Pi*((n+1)/x-1/2)))/Pi+(((n+2)-1)/n+1/2-arctan(tan(Pi*((n+1)/x-1/2)))/Pi-((n)/x+1/2-arctan(tan(Pi*((n)/x-1/2)))/Pi))*(x/2-x*arctan(tan(Pi*((n)/x-1/2)))/Pi)*x-1))/x.

⌊n/x⌋+1=(n+x-(x/2+x*arctan(tan(Pi*((n+1)/x-1/2)))/Pi+(((n+2)-1)/n+1/2-arctan(tan(Pi*((n+1)/x-1/2)))/Pi-((n)/x+1/2-arctan(tan(Pi*((n)/x-1/2)))/Pi))*(x/2-x*arctan(tan(Pi*((n)/x-1/2)))/Pi)*x-1))/x.

⌊n/x+1⌋=(n-((x+1)/2+(x+1)*arctan(tan(Pi*((n+1)/(x+1)-1/2)))/Pi+(((n+1+1)-1)/(x+1)+1/2-arctan(tan(Pi*((n+1)/(x+1)-1/2)))/Pi-((n)/(x+1)+1/2-arctan(tan(Pi*((n)/(x+1)-1/2)))/Pi))*((x+1)/2-(x+1)*arctan(tan(Pi*((n)/(x+1)-1/2)))/Pi)*(x+1)-1))/(x+1).

⌊x/n⌋=(x+n-(n/2+n*arctan(tan(Pi*(x/n-1/2)))/Pi+((x-1)/n+1/2-arctan(tan(Pi*((x-1)/n-1/2)))/Pi-((x-1)/n+1/2-arctan(tan(Pi*((x-1)/n-1/2)))/Pi))*(n/2-n*arctan(tan(Pi*((x-1)/n-1/2)))/Pi)*n))/n-(1-(1-cos(1/2*Pi)^(n-1))).

⌈n/x⌉=(x+n-(x/2+x*arctan(tan(Pi*(n/x-1/2)))/Pi+((n+1-1)/x+1/2-arctan(tan(Pi*((n+1-1)/x-1/2)))/Pi-((n-1)/x+1/2-arctan(tan(Pi*((n-1)/x-1/2)))/Pi))*(x/2-x*arctan(tan(Pi*((n-1)/x-1/2)))/Pi)*x))/x-(1-(1-cos(1/2*Pi)^(x-1))).

⌈n/x⌉-1=(n+x-(x/2+x*arctan(tan(pi*(n/x-1/2)))/pi+((n)/x+1/2-arctan(tan(pi*((n)/x-1/2)))/pi-((n-1)/x+1/2-arctan(tan(pi*((n-1)/x-1/2)))/pi))*(x/2-b*arctan(tan(pi*((n-1)/x-1/2)))/pi)*x))/x-1.

⌈n/x-1⌉=(n-(x/2+x*arctan(tan(pi*(n/x-1/2)))/pi+((n)/x+1/2-arctan(tan(pi*((n)/x-1/2)))/pi-((n-1)/x+1/2-arctan(tan(pi*((n-1)/x-1/2)))/pi))*(x/2-x*arctan(tan(pi*((n-1)/b-1/2)))/pi)*x))/x.

⌈x/n⌉=(x+n-(n/2+n*arctan(tan(Pi*(x/n-1/2)))/Pi+((x-1)/n+1/2-arctan(tan(Pi*((x-1)/n-1/2)))/Pi-((x-1)/n+1/2-arctan(tan(Pi*((x-1)/n-1/2)))/Pi))*(n/2-n*arctan(tan(Pi*((x-1)/n-1/2)))/Pi)*n))/n-(1-(1-cos(1/2*Pi)^(n-1))).