Sindbad~EG File Manager
/* Initialize menu text for MATHPERT */
/* Translator: translate text enclosed in quotation marks,
but do NOT translate text (usually formulas)
enclosed in dollar signs. Use the ISO-Latin1
character set.
8.6.98 last modified before translation
2.11.99 saved as .c file from the translator's .doc file
and went over the translations and put in the correct symbols,
as the files I received were in Unicode, hence all the symbols
were mangled.
6.22.99 translation complete
6.27.99 last modified before version 1.32
10.26.99 corrected some items in improper_integrals
12.30.99 added closing brackets in logarithmic_limits
1.4.00 added four more complex_hyperbolic operations
1.12.00 added missing commas in complex_hyperbolic
2.27.00-3.4.00 added text for series_convergence2
4.10.00 corrected improper_integrals text
7.10.00 deleted a line under special_limits
Added a missing parenthesis under advanced_limits
6.16.04 added "$$integral(u,t,a,a) = 0$$" in definite integration
6.21.04 modified text for complexroot and complexsqrt under complex_functions menu
1.27.06 four more operations under sg_function2 and 2 corrections under sg_function1
1.14.11 six more operations under inverse_hyperbolic
5.3.13 changed names of exported functions
5.17.13 added series_bernoulli
5.24.13 modified series_bernoulli
6.11.13 four more in series_bernoulli
6.12.13 added a missing paren
8.20.13 corrected the sign on six operations under series_ln
10.8.24 one more under integrate_by_substitition
*/
#include "mtext.h"
#include "operator.h"
#include "english1.h"
#include "lang.h"
static const char arithstr[] = "arithmétique"; /* save space with ONE copy of this */
static const char *mtext2[MAXMENUS][MAXLENGTH] =
{
{ /* double_angle */
"$sin 2\\theta = 2 sin \\theta cos \\theta $",
"$cos 2\\theta = cos^2 \\theta - sin^2 \\theta $",
"$cos 2\\theta = 1 - 2 sin^2 \\theta $",
"$cos 2\\theta = 2 cos^2 \\theta - 1$",
"$cos 2\\theta + 1 = 2cos^2 \\theta $",
"$cos 2\\theta - 1 = - 2 sin^2 \\theta $",
"$tan 2\\theta = 2 tan \\theta /(1 - tan^2 \\theta )$",
"$cot 2\\theta = (cot^2 \\theta -1) / (2 cot \\theta )$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$2 sin \\theta cos \\theta = sin 2\\theta $",
"$cos^2 \\theta - sin^2 \\theta = cos 2\\theta $",
"$1 - 2 sin^2 \\theta = cos 2\\theta $",
"$2 cos^2 \\theta - 1 = cos 2\\theta $"
},
{ /* multiple_angles */
"$n\\theta = (n-1)\\theta + \\theta $",
"$n\\theta = ?\\theta +(n-?)\\theta $",
"$sin 3\\theta = 3 sin \\theta - 4 sin^3 \\theta $",
"$cos 3\\theta = -3 cos \\theta + 4 cos^3 \\theta $",
"Développement de $sin n\\theta $ en $sin \\theta$, $cos \\theta $",
"Développement de $cos n\\theta $ en $sin \\theta$, $cos \\theta $"
},
{ /* verify_identities */
"Multiplication en croix",
"Permutation des deux membres",
"Déplacement de ? de gauche à droite",
"Déplacement de ? de droite à gauche",
"Addition de ? aux deux membres",
"Soustraction de ? des deux membres",
"Multiplication des deux membres par ?",
"Simplification d'un terme présent dans les deux membres",
"Elévation des deux membres à une même puissance",
"Composition des deux membres par la fonction racine carrée",
"Composition des deux membres par une fonction racine",
"Composition des deux membres par une même fonction",
arithstr,
"Vérification numérique",
"Procêde à un changement de variable de la forme u = ?",
},
{ /* solve_by_30_60_90 */
"$sin(u)=1/2$ si et seulement si $u=\\pi /6$ ou $5\\pi /6+2n\\pi $",
"$sin(u)=-1/2$ si et seulement si $u=-\\pi /6$ ou $-5\\pi /6+2n\\pi $",
"$sin(u)=\\sqrt 3/2$ si et seulement si $u=\\pi /3$ ou $2\\pi /3+2n\\pi $",
"$sin(u)=-\\sqrt 3/2$ si et seulement si $4u=-\\pi /3$ ou $-2\\pi /3+2n\\pi $",
"$cos(u)=\\sqrt 3/2$ si et seulement si $u=\\pm \\pi /6 + 2n\\pi $",
"$cos(u)=-\\sqrt 3/2$ si et seulement si $u=\\pm 5\\pi /6 + 2n\\pi $",
"$cos(u)=1/2$ si et seulement si $u=\\pm \\pi /3+2n\\pi $",
"$cos(u)=-1/2$ si et seulement si $u=\\pm 2\\pi /3+2n\\pi $",
"$tan(u)=1/\\sqrt 3$ si et seulement si $u= \\pi /6 + n\\pi $",
"$tan(u)=-1/\\sqrt 3$ si et seulement si $u= -\\pi /6 + n\\pi $",
"$tan(u)=\\sqrt 3$ si et seulement si $u= \\pi /3 + n\\pi $",
"$tan(u)=-\\sqrt 3$ si et seulement si $u= 2\\pi /3 + n\\pi $"
},
{ /* solve_by_45_45_90 */
"$sin u = 1/\\sqrt 2$ si $u=\\pi /4$ ou $3\\pi /4 + 2n\\pi $",
"$sin u=-1/\\sqrt 2$ si $u=5\\pi /4$ ou $7\\pi /4 + 2n\\pi $2",
"$cos u = 1/\\sqrt 2$ si $u=\\pi /4$ ou $7\\pi /4 + 2n\\pi $",
"$cos u=-1/\\sqrt 2$ si $u=3\\pi /4$ ou $5\\pi /4 + 2n\\pi $",
"tan u = 1 si $u= \\pi /4$ ou $5\\pi /4 + 2n\\pi $",
"tan u = -1 si $u=3\\pi /4$ ou $7\\pi /4 + 2n\\pi $"
},
{ /* zeroes_of_trig_functions */
"sin u = 0 si et seulement si $u = n\\pi $",
"sin u = 1 si et seulement si $u = \\pi /2+2n\\pi $",
"sin u = -1 si et seulement si $u = 3\\pi /2+2n\\pi $",
"cos u = 0 si et seulement si $u = (2n+1)\\pi /2$",
"cos u = 1 si et seulement si $u = 2n\\pi $",
"cos u = -1 si et seulement si $u = (2n+1)\\pi $",
"tan u = 0 si et seulement si sin u = 0",
"cot u = 0 si et seulement si cos u = 0"
},
{ /* inverse_trig_functions */
"sin u=c si et seulement si $u= (-1)^narcsin c+n\\pi $",
"sin u=c si et seulement si $u=arcsin(c)+2n\\pi $ or $2n\\pi +\\pi -arcsin(c)$",
"cos u=c si et seulement si $u=\\pm arccos c+2n\\pi $",
"tan u=c si et seulement si $u=arctan c+n\\pi $", /* c not ñ i */
"Calcul exact du l'arcsin",
"Calcul exact de l'arccos",
"Calcul exact de l'arctan",
"arccot x = arctan (1/x)",
"arcsec x = arccos (1/x)",
"arccsc x = arcsin (1/x)",
"arcsin(-x) = -arcsin x",
"$arccos(-x) = \\pi -arccos x$",
"arctan(-x) = -arctan x",
"Exprime les solutions sous forme périodique",
"Si |c|>1, il n'existe pas de u tel que sin u = c",
"Si |c|>1, il n'existe pas de u tel que cos u = c"
},
{ /* invsimp */
"$tan(arcsin x) = x/\\sqrt (1-x^2)$",
"$tan(arccos x) = \\sqrt (1-x^2)/x$",
"tan(arctan x) = x",
"sin(arcsin x) = x",
"$sin(arccos x) = \\sqrt (1-x^2)$",
"$sin(arctan x) = x/\\sqrt (x^2+1)$",
"$cos(arcsin x) = \\sqrt (1-x^2)$",
"cos(arccos x) = x",
"$cos(arctan x) = 1/\\sqrt (x^2+1)$",
"$sec(arcsin x) = 1/\\sqrt (1-x^2)$",
"$sec(arccos x) = 1/x$",
"$sec(arctan x) = \\sqrt (x^2+1)$",
"$arctan(tan \\theta ) = \\theta $6 si $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arcsin(sin \\theta ) = \\theta $ si $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arccos(cos \\theta ) = \\theta $ si $0\\le \\theta \\le \\pi $",
"arctan(tan x) = x + c1"
},
{ /* adding_arctrig_functions */
"arcsin x + arccos x = $\\pi /2$",
"$arctan x + arctan 1/x = \\pi x/2|x|$",
#if 0 /* Perhaps add these later */
"$arcsin x \\pm arcsin y = arcsin[x\\sqrt (1-y^2)\\pm y\\sqrt (1-x^2)]$",
"$arccos x + arccos y = arccos[xy-\\sqrt ((1-x^2)(1-y^2))]$",
"$arccos x - arccos y = arccos[xy+\\sqrt ((1-x^2)(1-y^2))]$",
"$arctan x + arctan y = arctan[(x+y)/(1-xy)]$",
"$arctan x - arctan y = arctan[(x-y)/(1+xy)]$",
#endif
},
{ /* complementary_trig */
"$sin(\\pi /2-\\theta ) = cos \\theta $",
"$cos(\\pi /2-\\theta ) = sin \\theta $",
"$tan(\\pi /2-\\theta ) = cot \\theta $",
"$cot(\\pi /2-\\theta ) = tan \\theta $",
"$sec(\\pi /2-\\theta ) = csc \\theta $",
"$csc(\\pi /2-\\theta ) = sec \\theta $",
"$sin \\theta = cos(\\pi /2-\\theta )$",
"$cos \\theta = sin(\\pi /2-\\theta )$",
"$tan \\theta = cot(\\pi /2-\\theta )$",
"$cot \\theta = tan(\\pi /2-\\theta )$",
"$sec \\theta = csc(\\pi /2-\\theta )$",
"$csc \\theta = sec(\\pi /2-\\theta )$"
},
{ /* complementary_degrees */
"$sin(90\\deg -\\theta ) = cos \\theta $",
"$cos(90\\deg -\\theta ) = sin \\theta $",
"$tan(90\\deg -\\theta ) = cot \\theta $",
"$cot(90\\deg -\\theta ) = tan \\theta $",
"$sec(90\\deg -\\theta ) = csc \\theta $",
"$csc(90\\deg -\\theta ) = sec \\theta $",
"$sin \\theta = cos(90\\deg -\\theta )$",
"$cos \\theta = sin(90\\deg -\\theta )$",
"$tan \\theta = cot(90\\deg -\\theta )$",
"$cot \\theta = tan(90\\deg -\\theta )$",
"$sec \\theta = csc(90\\deg -\\theta )$",
"$csc \\theta = sec(90\\deg -\\theta )$",
"$a\\deg + b\\deg = (a+b)\\deg $",
"$ca\\deg = (ca)\\deg $",
"$a\\deg /c = (a/c)\\deg $"
},
{ /* trig_odd_and_even */
"sin(-u) = - sin u",
"cos(-u) = cos u",
"tan(-u) = - tan u",
"cot(-u) = - cot u",
"sec(-u) = sec u",
"csc(-u) = - csc u",
"$sin^2(-u) = sin^2 u$",
"$cos^2(-u) = cos^2 u$",
"$tan^2(-u) = tan^2 u$",
"$cot^2(-u) = cot^2 u$",
"$sec^2(-u) = sec^2 u$",
"$csc^2(-u) = csc^2 u$"
},
{ /* trig_periodic */
"$sin(u+2\\pi ) = sin u$",
"$cos(u+2\\pi ) = cos u$",
"$tan(u+\\pi ) = tan u$",
"$sec(u+2\\pi ) = sec u$",
"$csc(u+2\\pi ) = csc u$",
"$cot(u+\\pi ) = cot u$",
"$sin^2(u+\\pi ) = sin^2 u$",
"$cos^2(u+\\pi ) = cos^2 u$",
"$sec^2(u+\\pi ) = sec^2 u$",
"$csc^2(u+\\pi ) = csc^2 u$",
"$sin u = -sin(u-\\pi )$",
"$sin u = sin(\\pi -u)$",
"$cos u = -cos(u-\\pi )$",
"$cos u = -cos(\\pi -u)$"
},
{ /* half_angle_identities */
"$sin^2(\\theta /2) = (1-cos \\theta )/2$",
"$cos^2(\\theta /2) = (1+cos \\theta )/2$",
"$sin^2(\\theta ) = (1-cos 2\\theta )/2$",
"$cos^2(\\theta ) = (1+cos 2\\theta )/2$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$tan(\\theta /2) = (sin \\theta )/(1+cos \\theta )$",
"$tan(\\theta /2) = (1-cos \\theta )/sin \\theta $",
"$cot(\\theta /2) = (1+cos \\theta )/(sin \\theta )$",
"$cot(\\theta /2) = sin \\theta /(1-cos \\theta )$",
"$sin(\\theta /2) = \\sqrt ((1-cos \\theta )/2)$ si $sin(\\theta /2)\\ge 0$",
"$sin(\\theta /2) = -\\sqrt ((1-cos \\theta )/2)$ si $sin(\\theta /2)\\le 0$",
"$cos(\\theta /2) = \\sqrt ((1+cos \\theta )/2)$ si $cos(\\theta /2)\\ge 0$",
"$cos(\\theta /2) = -\\sqrt ((1+cos \\theta )/2)$ si $cos(\\theta /2)\\le 0$",
"$\\theta = 2(\\theta /2)$"
},
{ /* product_and_factor_identities */
"$sin x cos x = \\onehalf sin 2x$",
"$sin x cos y = \\onehalf [sin(x+y)+sin(x-y)]$",
"$cos x sin y = \\onehalf [sin(x+y)-sin(x-y)]$",
"$sin x sin y = \\onehalf [cos(x-y)-cos(x+y)]$",
"$cos x cos y = \\onehalf [cos(x+y)+cos(x-y)]$",
"$sin x + sin y = 2 sin \\onehalf (x+y) cos \\onehalf (x-y)$",
"$sin x - sin y = 2 sin \\onehalf (x-y) cos \\onehalf (x+y)$",
"$cos x + cos y = 2 cos \\onehalf (x+y) cos \\onehalf (x-y)$",
"$cos x - cos y = -2 sin \\onehalf (x+y) sin \\onehalf (x-y)$",
"Remplacer $u,v$ dans fonctions trig."
},
{ /* limits */
"Expérimentation numérique",
"$lim u\\pm v = lim u \\pm lim v$",
"$lim u-v = lim u - lim v$",
"$$lim(t->a,c) = c$$ (c constante)",
"$$lim(t->a,t) = a$$",
"lim cu=c lim u (c constante)",
"lim -u = -lim u",
"lim uv = lim u lim v",
"$lim u^n = (lim u)^n$",
"lim c^v=c^(\\lim v) (c constante > 0)",
"lim u^v=(lim u)^(\\lim v)",
"$lim \\sqrt u=\\sqrt (lim u)$ si lim u>0",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ si n est impair",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ si lim u > 0",
"$$lim(t->a,f(t))=f(a)$$ (f polynôme)",
"lim |u| = |lim u|"
},
{ /* limits_of_quotients */
"lim cu/v = c lim u/v (c const)",
"lim c/v = c/lim v (c const)",
"lim u/v = lim u/lim v",
"Mise en facteur de (x-a)^n dans l'étude de la limite lorsque x tend vers a",
"Limite d'une fonction rationnelle",
"$a^n/b^n = (a/b)^n$",
"Rationalisation de la fonction",
"Séparation des termes ayant une limite finie non nulle", /* lim uv = lim u lim v where lim u is finite nonzero */
"Mise en facteur des constantes",
"Multiplication du numérateur et du dénominateur par ?",
"Division du numérateur et du dénominateur par ?",
"lim u/v = lim (u/?) / lim (v/?)",
"(ab+ac+d)/q = a(b+c)/q + d/q", /* limapartandfactor */
/* example : (sin x cos h + cos x sin h - sin x)/h */
},
{ /* quotients_of_roots */
"$\\sqrt a/b = \\sqrt (a/b^2)$ si b>0",
"$\\sqrt a/b = -\\sqrt (a/b^2)$ si b<0",
"$^n\\sqrt a/b = ^n\\sqrt (a/b^n)$ (b>0 ou n impair)",
"$^n\\sqrt a/b = -^n\\sqrt (a/b^n)$ (b<0, n pair)",
"$a/\\sqrt b = \\sqrt (a^2/b)$ si $a\\ge 0$",
"$a/\\sqrt b = -\\sqrt (a^2/b)$ si $a\\le 0$",
"$a/^n\\sqrt b = ^n\\sqrt (a^n/b)$ ($a\\ge 0$ ou n impair)",
"$a/^n\\sqrt b = -^n\\sqrt (a^n/b)$ ($a\\le 0$, n pair)"
},
{ /* lhopitalmenu */
"Rêgle de l'Hospital",
"Évaluation de la dérivée en une seule étape",
"lim u ln v = lim (ln v)/(1/u)",
"$lim u (ln v)^n = lim (ln v)^n/(1/u)$",
"$lim x^(-n) u = lim u/x^n$",
"lim u e^x = lim u/e^(-x)",
"Déplacement des fonctions trigonométriques au dénominateur",
"lim ?v = lim v/(1/?)",
"Mise au même dénominateur et simplification du numérateur"
},
{ /* special_limits */
"(sin t)/t \\to 1 lorsque t\\to 0",
"(tan t)/t \\to 1 lorsque t\\to 0",
"(1-cos t)/t \\to 0 lorsque t\\to 0",
"$(1-cos t)/t^2\\to \\onehalf $ lorsque t\\to 0",
"$$lim(t->0,(1+t)^(1/t)) = e$$",
"$(ln(1\\pm t))/t \\to \\pm 1$ lorsque t\\to 0",
"(e^t-1)/t \\to 1 lorsque t\\to 0",
"(e^(-t)-1)/t \\to -1 lorsque t\\to 0",
"$lim(t\\to 0,t^nln |t|)=0 (n > 0)$",
"lim(t\\to 0,cos(1/t)) n'existe pas",
"lim(t\\to 0,sin(1/t)) n'existe pas",
"lim(t\\to 0,tan(1/t)) n'existe pas",
"lim(t\\to \\pm \\infty cos t) n'existe pas",
"lim(t\\to \\pm \\infty sin t) n'existe pas",
"lim(t\\to \\pm \\infty tan t) n'existe pas"
},
{ /* hyper_limits */
"(sinh t)/t \\to 1 lorsque t\\to 0",
"(tanh t)/t \\to 1 lorsque t\\to 0",
"(cosh t - 1)/t \\to 0 lorsque t\\to 0",
"(cosh t - 1)/t^2\\to 1/2 lorsque t\\to 0",
},
{ /* advanced_limits */
"lim ln u=ln lim u (si lim u > 0)",
"Si f est continue, lim f(u)=f(lim u)",
"Changement de variable dans la limite", /* lim(t tend vers a,f(g(t)))=lim(u tend vers g(a),f(u)) */
"Calcul de la limite en une seule étape",
"$$lim(t->a, u^v) = lim(t->a, e^(v ln u))$$",
"lim ?v = lim v/(1/?)",
"Domaine ne permettant pas l'existence de la limite",
"$$lim(t->a,u) = e^(lim(t->a, ln u))$$",
"Théorême d'absorption: uv\\to 0 if v\\to 0 et $|u|\\le c$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/(\\sqrt u+v)$",
"lim u/v = limit des termes dominants",
"Terme dominant: lim(u+a) = lim(u) si a/u\\to 0",
"Remplacement de la somme par son terme dominant",
"f(non-défini) = non-défini",
"$$lim(t->a,e^u) = e^(lim(t->a, u))$$",
"lim(ln u) = ln(lim u)"
},
{ /* logarithmic_limits */
"$$lim(t->0+,t ln t) = 0$$",
"$$lim(t->0+,t^n ln t) = 0$$ si $n\\ge 1$",
"$$lim(t->0+,t (ln t)^n) = 0$$ si $n\\ge 1$",
"$$lim(t->0+,t^k (ln t)^n) = 0$$ si $k,n\\ge 1$",
"$$lim(t->infinity ,ln(t)/t) = 0$$",
"$$lim(t->infinity ,ln(t)^n/t) = 0$$ si $n\\ge 1$",
"$$lim(t->infinity ,ln(t)/t^n) = 0$$ si $n\\ge 1$",
"$$lim(t->infinity ,ln(t)^k/t^n) = 0$$ si $k,n\\ge 1$",
"$$lim(t->infinity ,t/ln(t)) = infinity $$",
"$$lim(t->infinity ,t/ln(t)^n) = infinity$$ si $n\\ge 1$",
"$$lim(t->infinity ,t^n/ln(t)) = infinity$$ si $n\\ge 1$",
"$$lim(t->infinity ,t^n/ln(t)^k) = infinity$$ si $k,n\\ge 1$"
},
{ /* limits_at_infinity */
"$$lim(t->infinity ,1/t^n) = 0$$ si $n\\ge 1$",
"$$lim(t->infinity,t^n) = infinity$$ si $n\\ge 1$",
"$$lim(t->infinity,e^t) = infinity$$",
"$$lim(t->-infinity,e^t) = 0$$",
"$$lim(t->infinity,ln t) = infinity $$",
"$$lim(t->infinity,\\sqrt t) = infinity $$",
"$$lim(t->infinity,t^n\\sqrt t) = infinity $$",
"$lim(t\\to\\pm \\infty ,arctan t) = \\pm \\pi /2$",
"$$lim(t->infinity,arccot t) = 0$$",
"$$lim(t->-infinity,arccot t) = pi $$",
"$lim(t\\to\\pm \\infty ,tanh t) = \\pm 1$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/\\sqrt u+v)$",
"lim sin u = sin(lim u)",
"lim cos u = cos(lim u)",
"Transformer une limite à $\\infty $ en une limite à 0",
"lim u/v = limite des termes principaux"
},
{ /* infinite_limits */
"$$lim(u->0, 1/u^(2n)) = infinity $$",
"$lim(1/u^n)$ est indéfini si $u\\to""0$ et $n$ est impair",
"$$lim(t->a+,1/u^n) = infinity $$ si $u\\to""0$",
"$$lim(t->a-,1/u^n)=-infinity $$ si $u\\to""0$ et $n$ est impair",
"$lim u/v$ est indéfini si $lim v =0$ et $lim u \\neq 0$",
"$$lim(t-> 0+,ln t) = -infinity $$",
"$lim(t\\to(2n+1)\\pi /2\\pm ,tan t) = \\pm \\infty $",
"$lim(t\\to n\\pi \\pm ,cot t) = \\pm \\infty $",
"$lim(t\\to(2n+1)\\pi /2\\pm ,sec t) = \\pm \\infty $",
"$lim(t\\to n\\pi \\pm ,csc t) = \\pm \\infty $",
"$lim(uv) = lim(u/?) lim(?v)$",
"$lim(uv) = lim(?u) lim(v/?)$"
},
{ /* infinities */
"$\\pm \\infty $/(strictement positif) = $\\pm \\infty $",
"nonnul/$\\pm \\infty $ = 0",
"(strictement positif)$\\times \\pm \\infty = \\pm \\infty $",
"$\\pm \\infty \\times \\infty = \\pm \\infty $",
"$\\pm \\infty $ + fini = $\\pm \\infty $",
"$\\infty + \\infty = \\infty $",
"$$u^infty = infty $$ si $u > 1$",
"$$u^infty = 0$$ si $0 < u < 1$",
"$$u^(-infty ) = 0$$ si $u > 1$",
"$$u^(-infty ) = infty $$ si $0 < u < 1$",
"$\\infty ^n = infty $ si $n > 0$",
"$\\infty - \\infty =$ est une forme indéterminée"
},
{ /* zero_denom */
"$a/0+ = \\infty $ si a>0",
"$a/0- = -\\infty $ si a>0",
"a/0 est une forme indéterminée",
"$\\infty /0+ = \\infty $",
"$\\infty /0- = -\\infty $",
"$\\infty /0$ est une forme indéterminée",
"$\\infty /0^2 = \\infty $",
"$\\infty /0^2^n = \\infty $",
"$a/0^2 = \\infty $ si a > 0",
"$a/0^2 = -\\infty $ si a < 0",
"$a/0^2^n = \\infty $ si a > 0",
"$a/0^2^n = -\\infty $ si a < 0"
},
{ /* more_infinities */
"$ln \\infty = log \\infty = \\infty $",
"$\\sqrt \\infty = \\infty $",
"$^n\\sqrt \\infty = \\infty $",
"$arctan \\pm \\infty = \\pm \\pi /2$",
"$arccot \\infty = 0$",
"$arccot -\\infty = \\pi $",
"$arcsec \\pm \\infty = \\pi /2$",
"$arccsc \\pm \\infty = 0$",
"Les fonctions trigonométriques usuelles n'ont pas de limite en $+?$.",
"$cosh \\pm \\infty = \\infty $",
"$sinh \\pm \\infty = \\pm \\infty $",
"$tanh \\pm \\infty = \\pm 1$",
"$ln 0 = -\\infty $"
},
{ /* polynomial_derivs */
"Si c est une constante, dc/dx=0",
"dx/dx = 1",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx(cu)=c du/dx (c indep of x)",
"d/dx x^n = n x^(n-1)",
"Dérivation de polynôme",
"f'(x) = d/dx f(x)"
},
{ /* derivatives */
"$$diff(f,x) = lim(h->0,(f(x+h)-f(x))/h)$$",
"Dérivation de polynôme",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx (cu) = c du/dx (c constante)",
"d/dx (u/c)=(1/c)du/dx (c constante)",
"d/dx x^n = n x^(n-1)",
"d/dx (uv) = u (dv/dx) + v (du/dx)",
"d/dx (1/v) = -(dv/dx)/v^2",
"d/dx (u/v)=[v(du/dx)-u(dv/dx)]/v^2",
"$d/dx \\sqrt x = 1/(2\\sqrt x)$",
"$$diff(root(n,x),x)= diff( x^(1/n),x)$$",
"$$diff(c/x^n,x) = -nc/x^(n+1)$$",
"d/dx |x| = x/|x|",
"f'(x) = d/dx f(x)"
},
{ /* dif_trig */
"d/dx sin x = cos x",
"d/dx cos x = - sin x",
"d/dx tan x = sec^2 x",
"d/dx sec x = sec x tan x",
"d/dx cot x = - csc^2 x",
"d/dx csc x = - csc x cot x"
},
{ /* dif_explog */
"d/dx e^x = e^x",
"d/dx c^x = (ln c) c^x, c constante",
"$$diff(u^v,x)= diff( e^(v ln u),x)$$",
"d/dx ln x = 1/x",
"d/dx ln |x| = 1/x",
"dy/dx = y (d/dx) ln y",
"d/dx e^u = e^u du/dx",
"d/dx c^u=(ln c)c^u du/dx, c const",
"d/dx ln u = (1/u)(du/dx)",
"d/dx ln |u| = (1/u) du/dx",
"d/dx ln(cos x) = -tan x",
"d/dx ln(sin x) = cot x"
},
{ /* dif_inverse_trig */
"$d/dx arctan x = 1/(1+x^2)$",
"$d/dx arcsin x = 1/\\sqrt (1-x^2)$",
"$d/dx arccos x = -1/\\sqrt (1-x^2)$",
"$d/dx arccot x = -1/(1+x^2)$",
"$d/dx arcsec x = 1/(|x|\\sqrt (x^2-1))$",
"$d/dx arccsc x = -1/(|x|\\sqrt (x^2-1))$",
"$d/dx arctan u = (du/dx)/(1+u^2)$",
"$d/dx arcsin u = (du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccos u = -(du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccot u = -(du/dx)/(1+u^2)$",
"$d/dx arcsec u=(du/dx)/(|u|\\sqrt (u^2-1))$",
"$d/dx arccsc u=-(du/dx)/(|u|\\sqrt (u^2-1))$"
},
{ /* chain_rule */
"d/dx u^n = nu^(n-1) du/dx",
"$d/dx \\sqrt u = (du/dx)/(2\\sqrt u)$",
"d/dx sin u = (cos u) du/dx",
"d/dx cos u = -(sin u) du/dx",
"$d/dx tan u = (sec^2 u) du/dx$",
"d/dx sec u=(sec u tan u) du/dx",
"$d/dx cot u = -(csc^2 u) du/dx$",
"d/dx csc u=-(csc u cot u) du/dx",
"d/dx |u| = (u du/dx)/|u|",
"d/dx f(u) = f'(u) du/dx",
"changement de variable de la forme u = ?",
"Elimination d'une variable ayant été définie"
},
{ /* maxima_and_minima */
"Expérimentation numérique",
"Etude des points d'annulation de la dérivée",
"Etude des bornes de l'intervalle d'étude",
"Etude des points de non dérivabilité",
"Détermination de slimites de la fonction aux bornes de l'intervalle ",
"Rejet des points situés en dehors de l'intervalle d'étude",
"Etablissement d'une table donnant pour chaque point candidat la valeur décimale de la fonction en ce point",
"Etablissement d'une table donnant pour chaque point candidat la valeur exacte de la fonction en ce point",
"Choix de la borne supérieure",
"Choix d ela borne inférieure",
"Calcul d ela dérivée en une seule étape",
"Résolution d'une équation élémentaire",
"Détermination de la limite en une seule étape",
"Elimination de paramêtres entiers",
"La fonction est constante"
},
{ /* implicit_diff */
"Calcul de la dérivée",
"Simplification",
"Résolution d'une équation élémentaire"
},
{ /* related_rates */
"Différentiation de l'équation",
"Calcul de la dérivée en une seule étape",
"Elimination de la dérivée grâce à un changement de variable",
"Résolution d'une équation élémentaire"
},
{ /* simplify */
"Simplification des sommes et produits",
"Elimination des fractions composées",
"Mise au même dénominateur et simplification",
"Mise en facteur du terme commun",
"Factorisation de l'expression",
"Développement des produits et simplification", /* meaning either collect ou cancel ou both */
"Mise en évidence du facteur commun dans u/v",
"Résolution d'une équation élémentaire",
"Ecriture sous forme polynomiale (en ?)",
"Ecriture comme un polynôme",
"Remise à 1 du coefficient dominant",
"$x^(\\onehalf) = \\sqrt x$", /* backtosqrts */
"Conversion des exposants rationnels en racines",
"Conversion en racines des exposants rationnels"
},
{ /* higher_derivatives */
"u=v => du/dx = dv/dx",
"$d^2u/dx^2 = (d/dx)(du/dx)$",
"$d^nu/dx^n= d/dx d^(n-1)u/dx^(n-1)$",
"$d/dx du/dx = d^2u/dx^2$",
"$d/dx d^nu/dx^n = d^(n+1)/dx^(n+1)$",
"Évaluation d'une dérivée en une seule étape",
"Évaluation numérique en un point"
},
{ /* basic_integration */
"$\\int 1 dt = t$",
"$\\int c dt = ct$ (c constante)",
"$\\int t dt = t^2/2$",
"$\\int cu dt = c\\int u dt$ (c constante)",
"$\\int (-u)dt = -\\int u dt$",
"$\\int u+v dt = \\int u dt + \\int v dt$",
"$\\int u-v dt = \\int u dt - \\int v dt$",
"$\\int au\\pm bv dt = a\\int u dt \\pm b\\int v dt$",
"$\\int t^n dt=t^(n+1)/(n+1) (n \\ne -1)$",
"$\\int 1/t^(n+1) dt= -1/(nt^n) (n \\ne 0)$",
"Intégration ou primitivation de polynômes",
"$\\int (1/t) dt = ln |t|$",
"$\\int 1/(t\\pm a) dt = ln |t\\pm a|$",
"Développement des produits dans l'intégrande",
"Développement de $(a+b)^n$ dans l'intégrande",
"$\\int |t| dt = t|t|/2$"
},
{ /* trig_integration */
"$\\int sin t dt = -cos t$",
"$\\int cos t dt = sin t$",
"$\\int tan t dt = -ln |cos t|$",
"$\\int cot t dt = ln |sin t|$",
"$\\int sec t dt = ln |sec t + tan t|$",
"$\\int csc t dt = ln |csc t - cot t|$",
"$\\int sec^2 t dt = tan t$",
"$\\int csc^2 t dt = -cot t$",
"$\\int tan^2 t dt = tan t - t$",
"$\\int cot^2 t dt = -cot t - t$",
"$\\int sec t tan t dt = sec t$",
"$\\int csc t cot t dt = -csc t$"
},
{ /* trig_integration2 */
"$\\int sin ct dt = -(1/c) cos ct$",
"$\\int cos ct dt = (1/c) sin ct$",
"$\\int tan ct dt = -(1/c) ln |cos ct|$",
"$\\int cot ct dt = (1/c) ln |sin ct|$",
"$\\int sec ct dt = (1/c) ln |sec ct + tan ct|$",
"$\\int csc ct dt = (1/c) ln |csc ct - cot ct|$",
"$\\int sec^2 ct dt = (1/c) tan ct$",
"$\\int csc^2 ct dt = -(1/c) cot ct$",
"$\\int tan^2 ct dt = (1/c) tan ct - t$",
"$\\int cot^2 ct dt = -(1/c) cot ct - t$",
"$\\int sec ct tan ct dt = (1/c) sec ct$",
"$\\int csc ct cot ct dt = -(1/c) csc ct$"
},
{ /* integrate_exp */
"$\\int e^t dt = e^t$",
"$\\int e^ct dt =(1/c) e^(ct)$",
"$\\int e^(-t)dt = -e^(-t)$",
"$\\int e^(-ct)dt = -(1/c) e^(-ct)$",
"$$integral( e^(t/c),t) = c e^(t/c)$$",
"$\\int c^t dt = (1/ln c) c^t$",
"$$ integral(u^v,t) = integral (e^(v ln u),t)$$",
"$\\int ln t = t ln t - t$",
"$$integral(e^(-t^2),t) = sqrt(pi)/2 Erf(t)$$"
},
{ /* integrate_by_substitution */
"Choix de la fonction pour le changement de variable, u = ?",
"Chix par l'ordinateur de la fonction u utilisée dans le changement de variable",
"Dérivation de l'équation",
"Évaluation de la dérivée en une seule étape",
"Ré-affichage de l'intégrale",
"Intégrande = $f(u) \\times du/dx$",
"$\\int f(u) (du/dx) dx = \\int f(u) du$",
"Élimination d'une variable ayant été définie",
"Intégration par changement de variable (u = ?)",
"Intégration par changement de variable",
"Intégrer par substitution et afficher les étapes"
},
{ /* integrate_by_parts */
"$\\int u dv = uv - \\int v du (u = ?)$",
"$\\int u dv = uv - \\int v du$",
"Ligne courante désormais considérée comme ligne d'origine",
"Déplacement dans le membre de gauche de l'intégrale d'origine",
"Évaluation de la dérivée en une seule étape",
"Intégration par changement de variable (u = ?)",
"Intégration par changement de variable",
"Évaluation d'une intégrale simple"
},
{ /* fundamental_theorem */
"$$integral(f'(x),x,a,b)=f(b)-f(a)$$",
"$$diff(integral(f(t),t,a,x),x) = f(x)$$"
},
{ /* definite_integration */
"$$eval(f(t),t,a,b) = f(b) - f(a)$$",
"$$eval(ln f(t),t,a,b) = ln(f(b)/f(a))$$",
"$$integral(u,t,a,b) = - integral(u,t,b,a)$$",
"$$integral(u,t,a,b) + integral(u,t,b,c) = integral(u,t,a,c)$$",
"$$integral(u,t,a,c) = integral(u,t,a,?) + integral(u,t,?,c)$$",
"Coupe l'intégrale $\\int |f(t)| dt$ aux zéros de f",
"Calcul numérique de l'intégrale avec paramêtre",
"Calcul numérique de l'intégrale",
"$$integral(u,t,a,a) = 0$$"
},
{ /* improper_integrals */
"$$integral(u,x,a,infinity) = lim(t->infinity,integral(u,x,a,t))$$",
"$$integral(u,x,-infinity,b) = lim(t->-infinity,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->a+,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->b-,integral(u,x,a,t))$$",
"Lintégrande ne tend pas vers 0 en $\\infty $",
"Lintégrande ne tend pas vers 0 en $-\\infty $"
},
{ /* oddandeven */
"$$integral(u,t,-a,a) = 0$$ (u impair)",
"$$integral(u,t,-a,a) = 2 integral(u,t,0,a)$$ (u pair)"
},
{ /* trig_substitutions */
"$x = a sin \\theta pour \\sqrt (a^2-x^2)$",
"$x = a tan \\theta pour \\sqrt (a^2+x^2)$",
"$x = a sec \\theta pour \\sqrt (x^2-a^2)$",
"$x = a sinh \\theta pour \\sqrt (a^2+x^2)$",
"$x = a cosh \\theta pour \\sqrt (x^2-a^2)$",
"$x = a tanh \\theta pour \\sqrt (a^2-x^2)$",
"Définition de la fonction réciproque pour le changement de variable, x = ?",
"Calcul de la dérivée",
"Intégration élémentaire en une seule étape"
},
{ /* trigonometric_integrals */
"$sin^2 t = (1-cos 2t)/2$ dans l'intégrale",
"$cos^2 t = (1+cos 2t)/2$ dans l'intégrale",
"u=cos x aprês avoir utilisé $sin^2=1-cos^2$",
"u=sin x aprês avoir utilisé $cos^2=1-sin^2$",
"u=tan x aprês avoir utilisé $sec^2=1+tan^2$",
"u=cot x aprês avoir utilisé $csc^2=1+cot^2$",
"u=sec x aprês avoir utilisé $tan^2=sec^2-1$",
"u=csc x aprês avoir utilisé $cot^2=csc^2-1$",
"$tan^2 x = sec^2 x - 1$ dans l'intégrande",
"$2cot^2 x = csc^2 x - 1$ dans l'intégrande",
"Reduction de $\\int sec^n x dx$",
"Réduction de $\\int csc^n x dx$",
"u = tan(x/2) (Changement de variable de Weierstrass)"
},
{ /* trigrationalize */
"Multiplication du numérateur et du dénominateur par 1+cos x",
"Multiplication du numérateur et du dénominateur par 1-cos x",
"Multiplication du numérateur et du dénominateur par 1+sin x",
"Multiplication du numérateur et du dénominateur par 1-sin x",
"Multiplication du numérateur et du dénominateur par sin x+cos x",
"Multiplication du numérateur et du dénominateur par cos x-sin x"
},
{ /* integrate_rational*/
"Division polynomiale",
"Factorisation du dénominateur (si facile)",
"Mise en évidence du facteur commun dans u/v",
"Factorisation sans carrés",
"Factorisation numérique du polynôme",
"Décomposition en éléments simples",
"Forme canonique",
"$\\int 1/(ct\\pm b) dt = (1/c) ln |ct\\pm b|$",
"$\\int 1/(ct\\pm b)^(n+1) dt = -1/nc(ct\\pm b)^n$",
"$\\int 1/(t^2+a^2)dt=(1/a)arctan(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/a)arccoth(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/2a)ln|(t-a)/(t+a)|$",
"$\\int 1/(a^2-t^2)dt=(1/a)arctanh(t/a)$",
"$\\int 1/(a^2-t^2)dt=(1/2a)ln|(t+a)/(a-t)|$"
},
{ /* integrate_sqrtdenom */
"Forme canonique",
"$\\int 1/\\sqrt (a^2-t^2)dt = arcsin(t/a)$",
"$\\int 1/\\sqrt (t^2\\pm a^2)dt)=ln|t+\\sqrt (t^2\\pm a^2)|$",
"$\\int 1/(t\\sqrt (t^2-a^2))dt=(1/a)arccos(t/a)$",
"Changement de variable amenant à une fraction rationnelle"
},
{ /* integrate_arctrig */
"$\\int arcsin z dz = z arcsin z + \\sqrt (1-z^2)$",
"$\\int arccos z dz = z arccos z - \\sqrt (1-z^2)$",
"$\\int arctan z dz = z arctan z - (1/2)ln(1+z^2)$",
"$\\int arccot z dz = z arccot z + (1/2)ln(1+z^2)$",
"$\\int arccsc z dz = z arccsc z+ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arccsc z dz = z arccsc z-ln(z + \\sqrt (z^2-1)) (z<0)$",
"$\\int arcsec z dz = z arcsec z-ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arcsec z dz = z arcsec z+ln(z + \\sqrt (z^2-1)) (z<0)$"
},
{ /* simplify_calculus */
"Simplification",
"Élimination des fractions composées",
"Mise au même dénominateur et simplification",
"Mise en facteur du terme commun",
"Factorisation de l'expression (non entiêre)",
"Développement des produits et simplification", /* meaning either collect ou cancel ou both */
"Miseen évidence du facteur commun dans u/v",
"Résolution d'une équation simple",
"Évaluation d'une dérivée en une seule étape",
"Évaluation d'une limite en une seule étape",
"Modification de l'intégrale grâce à un changement de variable ",
"Intégration élémentaire en une seule étape",
"Absorption du nombre dans la constante de primitivation"
},
{ /* integrate_hyperbolic */
"$\\int sinh u du = cosh u$",
"$\\int cosh u du = sinh u$",
"$\\int tanh u du = ln cosh u$",
"$\\int coth u du = ln sinh u$",
"$\\int csch u du = ln tanh(u/2)$",
"$\\int sech u du = arctan (sinh u)$"
},
{ /* series_geom1 */
"$$1/(1-x) = sum(x^n,n,0,infinity)$$",
"$1/(1-x) = 1+x+x^2+...$",
"$1/(1-x) = 1+x+x^2+...x^n...$",
"$$1/(1+x) = sum((-1)^n x^n,n,0,infinity)$$",
"$1/(1+x) = 1-x+x^2+...$",
"$1/(1+x) = 1-x+x^2+...(-1)^nx^n...$",
"$$sum(x^n,n,0,infinity)=1/(1-x)$$",
"$1+x+x^2+... = 1/(1-x)$",
"$1+x+x^2+...x^n...= 1/(1-x)$",
"$$sum((-1)^n x^n,n,0,infinity) = 1/(1+x)$$",
"$1-x+x^2+... = 1/(1+x)$",
"$1-x+x^2+...(-1)^nx^n... = 1/(1+x)$"
},
{ /* series_geom2 */
"$$x/(1-x) = sum(x^n,n,1,infinity)$$",
"$x/(1-x) = x+x^2+x^3+...$",
"$x/(1-x) = x+x^2+...x^n...$",
"$$x/(1+x) = sum((-1)^(n+1) x^n,n,1,infinity)$$",
"$x/(1+x) = x-x^2+x^3+...$",
"$x/(1+x) = x-x^2+...(-1)^(n+1)x^n...$",
"$$sum(x^n,n,1,infinity)=x/(1-x)$$",
"$x+x^2+x^3+...=x/(1-x)$",
"$x+x^2+...x^n...=x/(1-x)$",
"$$sum((-1)^(n+1) x^n,n,1,infinity)=x/(1+x) $$",
"$x-x^2+x^3+...=x/(1+x) $",
"$x-x^2+...(-1)^(n+1)x^n...=x/(1+x) $"
},
{ /* series_geom3 */
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,-3)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,2)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,2)$$",
"$$sum(x^(kn),n,0,infinity)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,-3)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,2)=1/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,-3)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,2)=x^m/(1-x^k)$$"
},
{ /* series_geom4 */
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,-3)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,2)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,2)$$",
"$$sum((-1)^nx^(kn),n,0,infinity)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,-3)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,2)=1/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,-3)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,2)=x^m/(1+x^k)$$"
},
{ /* series_geom5 */
"$$x^k/(1-x) = sum(x^n,n,k,infinity)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,-3)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,2)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,-3)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,2)$$",
"$$sum(x^n,n,k,infinity) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,-3) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,2) = x^k/(1-x)$$",
"$$sum((-1)^nx^n,n,k,infinity) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,-3) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,2) = x^k/(1+x)$$"
},
{ /* series_ln */
"$$ln(1-x) = -sum(x^n/n,n,1,infinity)$$",
"$$ln(1-x) = -sum(x^n/n,n,1,infinity,-3)$$",
"$$ln(1-x) =- sum(x^n/n,n,1,infinity,2)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,-3)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,2)$$",
"$$sum(x^n/n,n,1,infinity) = -ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,-3)=-ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,2)=-ln(1-x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,-3)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,2)=ln(1+x)$$"
},
{ /* series_trig */
"$$ sin x = sum( (-1)^n x^(2n+1)/(2n+1)!,n,0,infinity)$$",
"$sin x = x-x^3/3!+x^5/5!+...$",
"$sin x = x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+...$",
"$$cos x = sum( (-1)^n x^(2n)/(2n)!,n,0,infinity)$$",
"$cos x = 1-\\onehalf x^2+x^4/4! + ...$",
"$cos x = 1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+...$",
"$$sum((-1)^n x^(2n+1)/(2n+1)!,n,0,infinity) = sin x$$",
"$x-x^3/3!+x^5/5!+... = sin x$",
"$x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+... = sin x$",
"$$sum( (-1)^n x^(2n)/(2n)!,n,0,infinity) = cos x$$",
"$1-\\onehalf x^2+x^4/4! + ... = cos x$",
"$1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+... = cos x$"
},
{ /* series_exp */
"$$e^x = sum(x^n/n!,n,0,infinity)$$",
"$e^x = 1+x+x^2/2!+...$",
"$e^x = 1+x+...+x^n/n!...$",
"$$sum(x^n/n!,n,0,infinity)= e^x$$",
"$1+x+x^2/2!+ x^3/3!+... = e^x$",
"$1+x+...+x^n/n!... = e^x$",
"$$e^(-x) = sum((-x)^n x^n/n!,n,0,infinity)$$",
"$e^(-x) = 1-x+x^2/2!+...$",
"$e^(-x) = 1-x+...(-1)^nx^n/n!...$",
"$$sum((-1)^nx^n/n!,n,0,infinity)= e^(-x)$$",
"$1-x+x^2/2!+ x^3/3!+... = e^(-x)$",
"$1-x+...+(-1)^nx^n/n!... = e^(-x)$"
},
{ /* series_atan */
"$$arctan x = sum(x^(2n+1)/(2n+1),n,0,infinity)$$",
"$arctan x = x -x^3/3 + x^5/5 ...$",
"$arctan x = x -x^3/3 +...+ x^(2n+1)/(2n+1)+...$",
"$$sum(x^(2n+1)/(2n+1),n,0,infinity) = arctan x$$",
"$x -x^3/3 + x^5/5 ...=arctan x$",
"$x -x^3/3 +...+ x^(2n+1)/(2n+1)+...=arctan x$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,-3)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,2)$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,-3)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,2)= (1+x)^alpha$$"
},
{ /* series_bernoulli */
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$",
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$",
"$$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$",
"$$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$",
"$$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,2)$$",
"$$sec x = sum( (-1)^n (eulernumber(2n))/((2n)!) x^(2n),n,1,infinity)$$",
"$$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$",
"$$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,2)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity,-3)$$",
"$$zeta(s) = sum(1/n^s,n,1,infinity,-2)$$",
"$$sum((-1)^n/n,n,1,infinity) = ln 2$$"
},
{ /* series_appearance */
"Ecriture de la série sou la forme $a_0 + a_1 + ...$",
"Ecriture de la série sous la forme $a_0 + a_1 + a_2 + ... $",
"Ecriture de la série à l'aide de ... et du terme général",
"Ecriture de la série à l'aide de la notation sigma",
"Ecriture d'un autre terme avant ...",
"Ecriture de ? termes suplémentaires avant ...",
"Ecriture des termes après calcul des factorielles.",
"Absence d'évaluation des factorielles dans les termes",
"Ecriture décimale des coefficients",
"Absence d'utilisation de l'écriture décimale pour les coefficients"
},
{ /* series_algebra */
"Série amalgamante",
"Multiplication de séries",
"Multiplication de séries entières",
"Division d'une série entière par un polynôme",
"Division d'un polynôme par une série entière",
"Division de séries entières",
"Carré d'une série",
"Carré d'une série entière",
"Ecriture de $(\\sum a_k x^k)^n$ comme une série",
"addition de séries",
"Soustraction de séries"
//"Ecriture de lexponentielle d'une série comme une série,
},
{ /* series_manipulations */
"Affichage des premiers termes",
"Abaissement de la borne inférieure par soustraction de termes",
"Addition de ? à la variable d'indice",
"Soustraction de ? de la variable d'indice",
"Changement de nom de la variable d'indexation",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"Dérivation terme à terme de la série entière",
"$\\sum du/dx = d/dx \\sum u$",
"Intégration terme à terme de la série entière",
"$\\sum \\int u dx = \\int \\sum u dx$",
"Calcul de la somme des tous premiers termes",
"$$u = integral(diff(u,x),x)$$",
"$$u = integral(diff(u,t),t,0,x) + u0$$",
"$$u = diff(integral(u,x),x)$$",
"Détermination de la constante de primitivation",
"$\\sum a_k = \\sum a_(2k) + \\sum a_(2k+1)$"
},
{ /* series_convergence_tests */
"$\\sum u$ diverge si u ne tend pas vers zéro",
"Rêgle de comparaison avec une intégrale",
"Rêgle de D'Alembert",
"Rêgle de Cauchy",
"Rêgle de comparaison pour convergence",
"Rêgle de comparaison pour divergence",
"Rêgle des équivalents",
"Rêgle de condensation de Cauchy",
"Fin du test de la divergence",
"Achêvement de la comparaison avec une intégrale",
"Achêvement de la mise en ouvre de la rêgle de D'Alembert",
"Achêvement de la mise en ouvre de la rêgle de Cauchy",
"Achêvement de la comparaison",
"Achêvement de la comparaison",
"Achêvement de la comparaison par équivalents",
"Achêvement de la mise en ouvre de la rêgle de condensation de Cauchy",
},
{ /* series_convergence2 */
"Resultat de la règle de comparaison.",
"Resultat de la règle de comparaison.",
"$$sum(1/k,k,1,infinity) = infinity$$",
"$$sum(1/k^2,k,1,infinity) = pi^2/6$$",
"$$sum(1/k^s,k,1,infinity) = zeta(s)$$",
"$$zeta(2k) = (2^(2k-1) abs(bernoulli(2k)) pi^(2k))/factorial(2k)$$"
},
{ /* complex_functions */
"$$ln(u+iv) = ln(re^(i theta ))$$",
"$$ln(re^(i theta ))=ln r + i theta$$ $(-\\pi <\\theta \\le \\pi )$",
"$ln i = i\\pi /2$",
"$ln(-1) = i\\pi $",
"$ln(-a) = ln a + i\\pi (a > 0)$",
"$$cos theta = (e^(i theta ) + e^(-i theta ))/2$$",
"$$sin theta = (e^(i theta ) - e^(-i theta ))/(2i)$$",
"$$sqrt(re^(i theta))=sqrt(r) e^(i theta/2)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$$root(n,re^(i theta))=root(n,r) e^(i theta/n)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$$e^(i theta ) = cos theta + i sin theta $$",
"$$e^(x+iy) = e^x cos y + i e^x sin y$$",
"$$e^(i pi ) = -1$$",
"$$e^(-ipi ) = -1$$",
"$$e^(2n pi i) = 1$$",
"$$e^((2n pi + theta )i) = e^(i theta )$$",
"$$u^v = e^(v ln u)$$"
},
{ /* complex_hyperbolic */
"sin(it) = i sinh t",
"cos(it) = cosh t",
"cosh(it) = cos t",
"sinh(it) = i sin t",
"tan(it) = i tanh t",
"cot(it) = -i coth t",
"tanh(it) = i tan t",
"coth(it) = -i cot t",
"cos t + i sin t = e^(it)",
"cos t - i sin t = e^(-it)",
"$$(e^(i theta ) + e^(-i theta ))/2 = cos theta $$",
"$$(e^(i theta ) - e^(-i theta ))/2i = sin theta $$",
"$$e^(i theta ) + e^(-i theta ) = 2 cos theta $$",
"$$e^(i theta ) - e^(-i theta ) = 2i sin theta $$"
},
{ /* hyperbolic_functions */
"cosh u = (e^u+e^(-u))/2",
"e^u + e^-u = 2 cosh u",
"sinh u = (e^u-e^(-u))/2",
"e^u-e^(-u) = 2 sinh u",
"[e^u + e^-u]/2 = cosh u",
"[e^u-e^(-u)]/2 = sinh u",
"cosh(-u) = cosh u",
"sinh(-u) = -sinh u",
"cosh u + sinh u = e^u",
"cosh u - sinh u = e^(-u)",
"cosh 0 = 1",
"sinh 0 = 0",
"e^x = cosh x + sinh x",
"e^(-x) = cosh x - sinh x"
},
{ /* hyperbolic2 */
"$sinh^2u + 1 = cosh^2 u$",
"$cosh^2 u - 1 = sinh^2u $",
"$cosh^2 u - sinh^2u = 1$",
"$cosh^2 u = sinh^2u + 1$",
"$sinh^2u = cosh^2 u - 1$",
"$1 - tan^2u = sech^2u$",
"$1 - sech^2u = tan^2u$"
},
{ /* more_hyperbolic */
"tanh u = sinh u / cosh u",
"sinh u / cosh u = tanh u",
"coth u = cosh u / sinh u",
"cosh u / sinh u = coth u",
"sech u = 1 / cosh u",
"1 / cosh u = sech u",
"csch u = 1 / sinh u",
"1 / sinh u = csch u",
"$tanh^2 u + sech^2 u = 1$",
"$tanh^2 u = 1 - sech^2 u$",
"$sech^2 u = 1 - tanh^2 u $",
"$sinh(u\\pm v)=sinh u cosh v \\pm cosh u sinh v$",
"$cosh(u\\pm v)=cosh u cosh v \\pm sinh u sinh v$",
"sinh 2u = 2 sinh u cosh u",
"$cosh 2u = cosh^2 u + sinh^2 u$",
"$tanh(ln u) = (1-u^2)/(1+u^2)$"
},
{ /* inverse_hyperbolic */
"$arcsinh x = ln(x + \\sqrt (x^2+1))$",
"$arccosh x = ln(x + \\sqrt (x^2-1))$",
"$arctanh x = (1/2) ln((1+x)/(1-x))$",
"$sinh(asinh x) = x$",
"$cosh(acosh x) = x$",
"$tanh(atanh x) = x$",
"$coth(acoth x) = x$",
"$sech(asech x) = x$",
"$csch(acsch x) = x$"
},
{ /* dif_hyperbolic */
"$d/du sinh u = cosh u$",
"$d/du cosh u = sinh u$",
"$d/du tanh u = sech^2 u$",
"$d/du coth u = -csch^2 u$",
"$d/du sech u = -sech u tanh u$",
"$d/du csch u = -csch u coth u$",
"$d/du ln sinh u = coth u$",
"$d/du ln cosh u = tanh u$"
},
{ /* dif_inversehyperbolic */
"$d/du arcsinh u = 1/\\sqrt (u^2+1)$",
"$d/du arccosh u = 1/\\sqrt (u^2-1)$",
"$d/du arctanh u = 1/(1-u^2)$",
"$d/du arccoth u = 1/(1-u^2)$",
"$d/du arcsech u= -1/(u\\sqrt (1-u^2))$",
"$d/du arccsch u= -1/(|u|\\sqrt (u^2+1))$"
},
{ /* sg_function1 */
"sgn(x) = 1 si x > 0", /* sgpos */
"sgn(x) = -1 si x < 0", /* sgneg */
"sgn(0) = 0", /* sgzero */
"sgn(-x) = -sgn(x)", /* sgodd */
"-sgn(x) = sgn(-x)", /* sgodd2 */
"sgn(x) = |x|/x (x non nul)", /* sgabs1 */
"sgn(x) = x/|x| (x non nul)", /* sgabs2 */
"abs(x) = x sgn(x)", /* abssg */
"$sgn(x)^(2n)? = 1$", /* also sg(x)^(even/odd) sgevenpower */
"sgn(x)^(2n+1) = sgn(x)", /* also sg(x)^odd/odd sgoddpower */
"1/sgn(x) = sgn(x)", /* sgrecip */
"d/dx sgn(u) = 0 (u non nul)", /* difsg */
"$\\int sgn(x) = x sgn(x)$", /* intsg */
"$\\int sgn(u)v dx = sgn(u)\\int v dx$ (u non-nul)", /* sgint */
"sgn(x) = 1 lorsque x > 0", /* sgassumepos */
"sgn(x) = -1 lorsque x < 0" /* sgassumeneg */
},
{ /* sg_function2 */
"sgn(au) = sgn(u) si a > 0",
"sgn(au) = -sgn(u) si a < 0",
"sgn(au/b) = sgn(u) si a/b > 0",
"sgn(au/b) = - sgn(u) si a/b < 0",
"sgn(x^(2n+1)) = sgn(x)",
"sg(1/u) = sg(u)",
"sg(c/u) = sg(u) si c > 0",
"u sg(u) = |u|",
"|u| sg(u) = u"
},
{ /* bessel_functions */
"$$diff(J(0,x),x) = -J(1,x)$$",
"$$diff(J(1,x),x) = J(0,x) - J(1,x)/x$$",
"$$diff(J(n,x),x)=J(n-1,x)-(n/x)J(n,x)$$",
"$$diff(Y(0,x),x) = -Y(1,x)$$",
"$$diff(Y(1,x),x) = Y(0,x) - Y(1,x)/x$$",
"$$diff( Y(n,x),x)=Y(n-1,x)-(n/x)Y(n,x)$$"
},
{ /* modified_bessel_functions */
"$$diff(I(0,x),x) = -I(1,x)$$",
"$$diff(I(1,x),x) = I0(x) - I1(x)/x$$",
"$$diff(I(n,x),x)=I(n-1,x)-(n/x)I(n,x)$$",
"$$diff( K(0,x),x) = -K1(x)$$",
"$$diff(K(1,x),x) = -K0(x) - K1(x)/x$$",
"$$diff(K(n,x),x)= -K(n-1,x)-(n/x)K(n,x)$$"
},
{ /* functions_menu -- user-defined functions */
"" /* definitions of user-defined functions appear here. */
},
{"Développement", /* automode_only, this menu never appears! */
"Multiplication si annulations" /* but model.c uses corresponding entries in optable */
},
{"Suppression des racines carrées" /* automode_only2, also never appears */
},
{"" /* automode_only3, also never appears */
}
};
/*_____________________________________________________________*/
const char ** French_menutext2(int i)
/* returns an array of strings for the nitems + i-th menu
where nitems is the number of menus in frmtext.c
*/
{ return (const char **) mtext2[i];
}
Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists