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
*/
#define ENGLISH_DLL
#include "export.h" /* do not translate this or the next 3 lines */
#include "mtext.h"
#include "operator.h"
#include "english1.h"
#include "lang.h"
static const char arithstr[] = "arithm�tique"; /* 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 $",
"D�veloppement de $sin n\\theta $ en $sin \\theta $, $cos \\theta $",
"D�veloppement de $cos n\\theta $ en $sin \\theta $, $cos \\theta $"
},
{ /* verify_identities */
"Multiplication en croix",
"Permutation des deux membres",
"D�placement de ? de gauche � droite",
"D�placement de ? de droite � gauche",
"Addition de ? aux deux membres",
"Soustraction de ? des deux membres",
"Multiplication des deux membres par ?",
"Simplification d'un terme pr�sent dans les deux membres",
"El�vation des deux membres � une m�me puissance",
"Composition des deux membres par la fonction racine carr�e",
"Composition des deux membres par une fonction racine",
"Composition des deux membres par une m�me fonction",
arithstr,
"V�rification num�rique",
"Proc�de � un changement de variable de la forme u = ?",
},
{ /* solve_by_30_60_90 */
"$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 $", /* c not � i */
"Calcul exact du l'arcsin",
"Calcul exact de l'arccos",
"Calcul exact de l'arctan",
"arccot x = arctan (1/x)",
"arcsec x = arccos (1/x)",
"arccsc x = arcsin (1/x)",
"arcsin(-x) = -arcsin x",
"$arccos(-x) = \\pi -arccos x$",
"arctan(-x) = -arctan x",
"Exprime les solutions sous forme p�riodique",
"Si |c|>1, il n'existe pas de u tel que sin u = c",
"Si |c|>1, il n'existe pas de u tel que cos u = c"
},
{ /* invsimp */
"$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) if sin(\\theta /2)\\ge 0$",
"$sin(\\theta /2) = -\\sqrt ((1-cos \\theta )/2) if sin(\\theta /2)\\le 0$",
"$cos(\\theta /2) = \\sqrt ((1+cos \\theta )/2) if cos(\\theta /2)\\ge 0$",
"$cos(\\theta /2) = -\\sqrt ((1+cos \\theta )/2) if 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)$",
"Remplacement de u et v par expressions trigonom�triques"
},
{ /* limits */
"Exp�rimentation num�rique",
"$lim u\\pm v = lim u \\pm lim v$",
"$lim u-v = lim u - lim v$",
"lim(t\32a,c) = c (c constante)",
"lim(t\32a,t) = a",
"lim cu=c lim u (c constante)",
"lim -u = -lim u",
"lim uv = lim u lim v",
"$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\32a,f(t))=f(a) (f polyn�me)",
"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",
"fMise en facteur de (x-a)^n dans l'�tude de la limite lorsque x tend vers a",
"Limite d'une fonction rationnelle",
"$a^n/b^n = (a/b)^n$",
"Rationalisation de la fonction",
"S�paration 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 num�rateur et du d�nominateur par ?",
"Division du num�rateur et du d�nominateur 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 */
"R�gle de l'Hospital",
"�valuation de la d�riv�e en une seule �tape",
"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)",
"D�placement des fonctions trigonom�triques au d�nominateur",
"lim ?v = lim v/(1/?)",
"Mise au m�me d�nominateur et simplification du num�rateur"
},
{ /* special_limits */
"(sin t)/t \32 1 lorsque t\32""0",
"(tan t)/t \32 1 lorsque t\32""0",
"(1-cos t)/t \32 0 lorsque t\32""0",
"$(1-cos t)/t^2\32""\\onehalf $ lorsque t\32""0",
"lim(t\32""0,(1+t)^(1/t)) = e",
"$(ln(1\\pm t))/t \32 \\pm 1$ lorsque t\32""0",
"(e^t-1)/t \32 1 lorsque t\32""0",
"(e^(-t)-1)/t \32 -1 lorsque t\32""0",
"$lim(t\32""0,t^nln |t|)=0 (n > 0)$",
"lim(t\32""0,cos(1/t)) n'existe pas",
"lim(t\32""0,sin(1/t)) n'existe pas",
"lim(t\32""0,tan(1/t)) n'existe pas",
"lim(t\32""$\\pm \\infty $,cos t) n'existe pas",
"lim(t\32""$\\pm \\infty $,sin t) n'existe pas",
"lim(t\32""$\\pm \\infty $,tan t) n'existe pas"
},
{ /* hyper_limits */
"(sinh t)/t \32 1 lorsque t\32""0",
"(tanh t)/t \32 1 lorsque t\32""0",
"(cosh t - 1)/t \32 0 lorsque t\32""0",
"(cosh t - 1)/t^2\32""1/2 lorsque t\32""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 �tape",
"lim u^v = lim e^(v ln u)",
"lim ?v = lim v/(1/?)",
"Domaine ne permettant pas l'existence de la limite",
"lim u = e^(lim ln u)",
"Th�or�me d'absorption: uv\32""0 if v\32""0 & $|u|\\le c$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/(\\sqrt u+v)$",
"lim u/v = limit des termes dominants",
"Terme dominant: lim(u+a) = lim(u) si a/u\32""0",
"Remplacement de la somme par son terme dominant",
"f(non-d�fini) = non-d�fini",
"lim(e^u) = e^(lim u)",
"lim(ln u) = ln(lim u)"
},
{ /* logarithmic_limits */
"$lim(t\32""0+,t ln t) = 0$",
"$lim(t\32""0+,t^n ln t) = 0 si n\\ge 1$",
"$lim(t\32""0+,t (ln t)^n) = 0 si n\\ge 1$",
"$lim(t\32""0+,t^k (ln t)^n) = 0 si k,n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t) = 0$",
"$lim(t\32\\infty ,ln(t)^n/t) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t^n) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,ln(t)^k/t^n) = 0 si k,n\\ge 1$",
"$lim(t\32\\infty ,t/ln(t)) = \\infty $",
"$lim(t\32\\infty ,t/ln(t)^n) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)^k) = \\infty si k,n\\ge 1$"
},
{ /* limits_at_infinity */
"$lim(t\32\\infty ,1/t^n) = 0 si n\\ge 1$",
"$lim(t\32\\infty ,t^n) = \\infty si n\\ge 1$",
"$lim(t\32\\infty ,e^t) = \\infty $",
"$lim(t\32-\\infty ,e^t) = 0$",
"$lim(t\32\\infty ,ln t) = \\infty $",
"$lim(t\32\\infty ,\\sqrt t) = \\infty $",
"$lim(t\32\\infty ,^n\\sqrt t) = \\infty $",
"$lim(t\32\\pm \\infty ,arctan t) = \\pm \\pi /2$",
"$lim(t\32\\infty ,arccot t) = 0$",
"$lim(t\32-\\infty ,arccot t) = \\pi $",
"$lim(t\32\\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)",
"Transformation d'une limite en $\\infty $ en une limite en 0",
"lim u/v = limite des termes dominants"
},
{ /* infinite_limits */
"$lim(1/u^2^n) = \\infty $ si u\32""0",
"lim(1/u^n) n'existe pas si u\32""0, n impair",
"lim(t\32a+,1/u^n) = $\\infty $ si u\32""0",
"lim(t\32a-,1/u^n)=-$\\infty $, u\32""0, n impair",
"lim u/v n'existe pas si lim v =0, lim u #0",
"lim(t\32""0+,ln t) = -$\\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,tan t) = \\pm \\infty $",
"$lim(t\32n\\pi \\pm ,cot t) = \\pm \\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,sec t) = \\pm \\infty $",
"$lim(t\32n\\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 $ if u > 1",
"$u^\\infty = 0$ if 0 < u < 1",
"$u^(-\\infty ) = 0$ if u > 1",
"$u^(-\\infty ) = \\infty $ if 0 < u < 1",
"$\\infty ^n = \\infty $ if n > 0",
"$\\infty - \\infty =$ est une forme ind�termin�e"
},
{ /* zero_denom */
"$a/0+ = \\infty $ si a>0",
"$a/0- = -\\infty $ si a>0",
"a/0 est une forme ind�termin�e",
"$\\infty /0+ = \\infty $",
"$\\infty /0- = -\\infty $",
"$\\infty /0$ est une forme ind�termin�e",
"$\\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 trigonom�triques 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)",
"D�rivation de polyn�me",
"f'(x) = d/dx f(x)"
},
{ /* derivatives */
"$$diff(f,x) = lim(h->0,(f(x+h)-f(x))/h)$$",
"D�rivation de polyn�me",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"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",
"$d/dx \\sqrt x = 1/(2\\sqrt x)$",
"$d/dx ^n\\sqrt x = d/dx x^(1/n)$",
"$d/dx (c/x^n) = -nc/x^(n+1)$",
"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",
"d/dx u^v= (d/dx) e^(v ln u)",
"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 �t� d�finie"
},
{ /* maxima_and_minima */
"Exp�rimentation num�rique",
"Etude des points d'annulation de la d�riv�e",
"Etude des bornes de l'intervalle d'�tude",
"Etude des points de non d�rivabilit�",
"D�termination de slimites de la fonction aux bornes de l'intervalle ",
"Rejet des points situ�s en dehors de l'intervalle d'�tude",
"Etablissement d'une table donnant pour chaque point candidat la valeur d�cimale 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 sup�rieure",
"Choix d ela borne inf�rieure",
"Calcul d ela d�riv�e en une seule �tape",
"R�solution d'une �quation �l�mentaire",
"D�termination de la limite en une seule �tape",
"Elimination de param�tres entiers",
"La fonction est constante"
},
{ /* implicit_diff */
"Calcul de la d�riv�e",
"Simplification",
"R�solution d'une �quation �l�mentaire"
},
{ /* related_rates */
"Diff�rentiation de l'�quation",
"Calcul de la d�riv�e en une seule �tape",
"Elimination de la d�riv�e gr�ce � un changement de variable",
"R�solution d'une �quation �l�mentaire"
},
{ /* simplify */
"Simplification des sommes et produits",
"Elimination des fractions compos�es",
"Mise au m�me d�nominateur et simplification",
"Mise en facteur du terme commun",
"Factorisation de l'expression",
"D�veloppement des produits et simplification", /* meaning either collect ou cancel ou both */
"Mise en �vidence du facteur commun dans u/v",
"R�solution d'une �quation �l�mentaire",
"Ecriture sous forme polynomiale (en ?)",
"Ecriture comme un polyn�me",
"Remise � 1 du coefficient dominant",
"$x^(1/2) = \\sqrt x$", /* backtosqrts */
"Conversion des exposants rationnels en racines",
"Conversion en racines des exposants rationnels"
},
{ /* higher_derivatives */
"u=v => du/dx = dv/dx",
"$d^2u/dx^2 = (d/dx)(du/dx)$",
"$d^nu/dx^n= d/dx d^(n-1)u/dx^(n-1)$",
"$d/dx du/dx = d^2u/dx^2$",
"$d/dx d^nu/dx^n = d^(n+1)/dx^(n+1)$",
"�valuation d'une d�riv�e en une seule �tape",
"�valuation num�rique 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)",
"$\\int (-u)dt = -\\int u dt$",
"$\\int u+v dt = \\int u dt + \\int v dt$",
"$\\int u-v dt = \\int u dt - \\int v dt$",
"$\\int au\\pm bv dt = a\\int u dt \\pm b\\int v dt$",
"$\\int t^n dt=t^(n+1)/(n+1) (n # -1)$",
"$\\int 1/t^(n+1) dt= -1/(nt^n) (n # 0)$",
"Int�gration ou primitivation de polyn�mes",
"$\\int (1/t) dt = ln |t|$",
"$\\int 1/(t\\pm a) dt = ln |t\\pm a|$",
"D�veloppement des produits dans l'int�grande",
"D�veloppement de $(a+b)^n$ dans l'int�grande",
"$\\int |t| dt = t|t|/2$"
},
{ /* 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 */
"$\\int e^t dt = e^t$",
"$\\int e^ct dt =(1/c) e^(ct)$",
"$\\int e^(-t)dt = -e^(-t)$",
"$\\int e^(-ct)dt = -(1/c) e^(-ct)$",
"$\\int e^(t/c)dt = c e^(t/c)$",
"$\\int c^t dt = (1/ln c) c^t$",
"$\\int u^v dt = \\int (e^(v ln u) dt$",
"$\\int ln t = t ln t - t$",
"$$integral(e^(-t^2),t) = sqrt(pi)/2 Erf(t)$$"
},
{ /* integrate_by_substitution */
"Choix de la fonction pour le changement de variable, u = ?",
"Chix par l'ordinateur de la fonction u utilis�e dans le changement de variable",
"D�rivation de l'�quation",
"�valuation de la d�riv�e en une seule �tape",
"R�-affichage de l'int�grale",
"Int�grande = $f(u) \\times du/dx$",
"$\\int f(u) (du/dx) dx = \\int f(u) du$",
"�limination d'une variable ayant �t� d�finie",
"Int�gration par changement de variable (u = ?)",
"Int�gration par changement de variable"
},
{ /* integrate_by_parts */
"$\\int u dv = uv - \\int v du (u = ?)$",
"$\\int u dv = uv - \\int v du$",
"Ligne courante d�sormais consid�r�e comme ligne d'origine",
"D�placement dans le membre de gauche de l'int�grale d'origine",
"�valuation de la d�riv�e en une seule �tape",
"Int�gration par changement de variable (u = ?)",
"Int�gration par changement de variable",
"�valuation d'une int�grale 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'int�grale $\\int |f(t)| dt$ aux z�ros de f",
"Calcul num�rique de l'int�grale avec param�tre",
"Calcul num�rique de l'int�grale",
"$$integral(u,t,a,a) = 0$$"
},
{ /* improper_integrals */
"$$integral(u,x,a,infinity) = lim(t->infinity,integral(u,x,a,t))$$",
"$$integral(u,x,-infinity,b) = lim(t->-infinity,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->a+,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->b-,integral(u,x,a,t))$$",
"L�int�grande ne tend pas vers 0 en $\\infty $",
"L�int�grande 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 */
"$x = a sin \\theta {pour \\sqrt (a^2-x^2)}$",
"$x = a tan \\theta {pour \\sqrt (a^2+x^2)}$",
"$x = a sec \\theta {pour \\sqrt (x^2-a^2)}$",
"$x = a sinh \\theta {pour \\sqrt (a^2+x^2)}$",
"$x = a cosh \\theta {pour \\sqrt (x^2-a^2)}$",
"$x = a tanh \\theta {pour \\sqrt (a^2-x^2)}$",
"D�finition de la fonction r�ciproque pour le changement de variable, x = ?",
"Calcul de la d�riv�e",
"Int�gration �l�mentaire en une seule �tape"
},
{ /* trigonometric_integrals */
"$sin^2 t = (1-cos 2t)/2$ dans l'int�grale",
"$cos^2 t = (1+cos 2t)/2$ dans l'int�grale",
"u=cos x apr�s avoir utilis� $sin^2=1-cos^2$",
"u=sin x apr�s avoir utilis� $cos^2=1-sin^2$",
"u=tan x apr�s avoir utilis� $sec^2=1+tan^2$",
"u=cot x apr�s avoir utilis� $csc^2=1+cot^2$",
"u=sec x apr�s avoir utilis� $tan^2=sec^2-1$",
"u=csc x apr�s avoir utilis� $cot^2=csc^2-1$",
"$tan^2 x = sec^2 x - 1$ dans l'int�grande",
"$2cot^2 x = csc^2 x - 1$ dans l'int�grande",
"Reduction de $\\int sec^n x dx$",
"R�duction de $\\int csc^n x dx$",
"u = tan(x/2) (Changement de variable de Weierstrass)"
},
{ /* trigrationalize */
"Multiplication du num�rateur et du d�nominateur par 1+cos x",
"Multiplication du num�rateur et du d�nominateur par 1-cos x",
"Multiplication du num�rateur et du d�nominateur par 1+sin x",
"Multiplication du num�rateur et du d�nominateur par 1-sin x",
"Multiplication du num�rateur et du d�nominateur par sin x+cos x",
"Multiplication du num�rateur et du d�nominateur par cos x-sin x"
},
{ /* integrate_rational*/
"Division polynomiale",
"Factorisation du d�nominateur (si facile)",
"Mise en �vidence du facteur commun dans u/v",
"Factorisation sans carr�s",
"Factorisation num�rique du polyn�me",
"D�composition en �l�ments 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 � 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",
"�limination des fractions compos�es",
"Mise au m�me d�nominateur et simplification",
"Mise en facteur du terme commun",
"Factorisation de l'expression (non enti�re)",
"D�veloppement des produits et simplification", /* meaning either collect ou cancel ou both */
"Miseen �vidence du facteur commun dans u/v",
"R�solution d'une �quation simple",
"�valuation d'une d�riv�e en une seule �tape",
"�valuation d'une limite en une seule �tape",
"Modification de l'int�grale gr�ce � un changement de variable ",
"Int�gration �l�mentaire en une seule �tape",
"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 s�rie sou la forme $a_ 0 + a_ 1 + ...$",
"Ecriture de la s�rie sous la forme $a_ 0 + a_ 1 + a_ 2 + ... $",
"Ecriture de la s�rie � l�aide de ... et du terme g�n�ral",
"Ecriture de la s�rie � l�aide de la notation sigma",
"Ecriture d�un autre terme avant ...",
"Ecriture de ? termes supl�mentaires avant ...",
"Ecriture des termes apr�s calcul des factorielles.",
"Absence d��valuation des factorielles dans les termes",
"Ecriture d�cimale des coefficients",
"Absence d�utilisation de l��criture d�cimale pour les coefficients"
},
{ /* series_algebra */
"S�rie amalgamante",
"Multiplication de s�ries",
"Multiplication de s�ries enti�res",
"Division d�une s�rie enti�re par un polyn�me",
"Division d�un polyn�me par une s�rie enti�re",
"Division de s�ries enti�res",
"Carr� d�une s�rie",
"Carr� d�une s�rie enti�re",
"Ecriture de $(\\sum a_k x^k)^n$ comme une s�rie",
"addition de s�ries",
"Soustraction de s�ries"
//"Ecriture de l�exponentielle d�une s�rie comme une s�rie�,
},
{ /* series_manipulations */
"Affichage des premiers termes",
"Abaissement de la borne inf�rieure par soustraction de termes",
"Addition de ? � la variable d�indice",
"Soustraction de ? de la variable d�indice",
"Changement de nom de la variable d�indexation",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"D�rivation terme � terme de la s�rie enti�re",
"$\\sum du/dx = d/dx \\sum u$",
"Int�gration terme � terme de la s�rie enti�re",
"$\\sum \\int u dx = \\int \\sum u dx$",
"Calcul de la somme des tous premiers termes",
"$$u = integral(diff(u,x),x)$$",
"$$u = integral(diff(u,t),t,0,x) + u0$$",
"$$u = diff(integral(u,x),x)$$",
"D�termination 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 z�ro",
"R�gle de comparaison avec une int�grale",
"R�gle de D'Alembert",
"R�gle de Cauchy",
"R�gle de comparaison pour convergence",
"R�gle de comparaison pour divergence",
"R�gle des �quivalents",
"R�gle de condensation de Cauchy",
"Fin du test de la divergence",
"Ach�vement de la comparaison avec une int�grale",
"Ach�vement de la mise en ouvre de la r�gle de D'Alembert",
"Ach�vement de la mise en ouvre de la r�gle de Cauchy",
"Ach�vement de la comparaison",
"Ach�vement de la comparaison",
"Ach�vement de la comparaison par �quivalents",
"Ach�vement de la mise en ouvre de la r�gle de condensation de Cauchy",
},
{ /* series_convergence2 */
"Resultat de la r�gle de comparaison.",
"Resultat de la r�gle 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^(-i\\pi ) = -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$", /* also sg(x)^(even/odd) sgevenpower */
"sgn(x)^(2n+1) = sgn(x)", /* also sg(x)^odd/odd sgoddpower */
"1/sgn(x) = sgn(x)", /* sgrecip */
"d/dx sgn(u) = 0 (u nonzero)", /* difsg */
"$\\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 */
"d/dx J0(x) = -J1(x)",
"d/dx J1(x) = J0(x) - J1(x)/x",
"d/dx J(n,x)=J(n-1,x)-(n/x)J(n,x)",
"d/dx Y0(x) = -Y1(x)",
"d/dx Y1(x) = Y0(x) - Y1(x)/x",
"d/dx Y(n,x)=Y(n-1,x)-(n/x)Y(n,x)"
},
{ /* modified_bessel_functions */
"d/dx I0(x) = -I1(x)",
"d/dx I1(x) = I0(x) - I1(x)/x",
"d/dx I(n,x)=I(n-1,x)-(n/x)I(n,x)",
"d/dx K0(x) = -K1(x)",
"d/dx K1(x) = -K0(x) - K1(x)/x",
"d/dx K(n,x)= -K(n-1,x)-(n/x)K(n,x)"
},
{ /* functions_menu -- user-defined functions */
"" /* definitions of user-defined functions appear here. */
},
{"D�veloppement", /* automode_only, this menu never appears! */
"Multiplication si annulations" /* but model.c uses corresponding entries in optable */
},
{"Suppression des racines carr�es" /* 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