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/*************** -*- Mode: MACSYMA; Package: MAXIMA -*- ******************/
/***************************************************************************
*** *****
*** Copyright (c) 1984 by William Schelter,University of Texas *****
*** All rights reserved *****
***************************************************************************/
kill(functions,arrays,values);
done$
ev(sin(x),exponentialize);
-%i*(%e^(%i*x)-%e^-(%i*x))/2$
taylor(sin(x)/x,x,0,4);
''(taylor(1-x^2/6+x^4/120,x,0,4))$
ev(cos(x)^2-sin(x)^2,sin(x)^2 = 1-cos(x)^2);
2*cos(x)^2-1$
(sqrt(-4)+sqrt(2.25))^2;
(2*%i+1.5)^2$
expand(%);
6.0*%i-1.75$
expand(sqrt(2*%i));
(-1)^(1/4)*sqrt(2)$
kill(all);
done$
sin(x)+cos(y)+(w+1)^2+'diff(sin(w),w);
cos(y)+sin(x)+'diff(sin(w),w,1)+(w+1)^2$
ev(%,sin,expand,diff,x = 2,y = 1);
cos(w)+w^2+2*w+cos(1)+sin(2)+1;
ev(x+y,x:a+y,y:2);
y+a+2$
'diff(y^2+x*y+x^2,x,2,y,1);
'diff(y^2+x*y+x^2,x,2,y,1)$
ev(%,diff);
0$
exp:2*x-3*y = 3;
2*x-3*y = 3$
-3*x+2*y = -4;
2*y-3*x = -4$
solve([exp,%]);
[[y = -1/5,x = 6/5]]$
ev(exp,%);
3 = 3$
x+1/x > gamma(1/2);
x+1/x > sqrt(%pi)$
ev(%,numer,x = 1/2);
2.5 > 1.772453850905516;
/* tops 20 :2.5 > 1.77245384$ */
ev(%,pred);
true$
zeroequiv(sin(2*x)-2*sin(x)*cos(x),x);
true$
zeroequiv(%e^x+x,x);
false$
zeroequiv(log(a*b)-log(a)-log(b),a);
dontknow$
(1/(x+y)^4-3/(y+z)^3)^2;
(1/(y+x)^4-3/(z+y)^3)^2$
expand(%,2,0);
-6/((y+x)^4*(z+y)^3)+9/(z+y)^6+1/(y+x)^8$
expand(a . (b+c . (d+e)+f));
a . f+a . c . e+a . c . d+a . b$
expand((x+1)^3);
x^3+3*x^2+3*x+1$
exp:(x+1)^7;
(x+1)^7$
expand(%);
x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$
expand(exp,7,7);
x^7+7*x^6+21*x^5+35*x^4+35*x^3+21*x^2+7*x+1$
ev(a*(b+c)+a*(b+c)^2,expop:1);
a*(c+b)^2+a*c+a*b$
ratexpand((2*x-3*y)^3);
-27*y^3+54*x*y^2-36*x^2*y+8*x^3$
exp:(x-1)/(x+1)^2+1/(x-1);
(x-1)/(x+1)^2+1/(x-1)$
expand(%);
x/(x^2+2*x+1)-1/(x^2+2*x+1)+1/(x-1)$
ratexpand(exp);
2*x^2/(x^3+x^2-x-1)+2/(x^3+x^2-x-1)$
sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2);
sin(x/(x^2+x)) = %e^((log(x)+1)^2-log(x)^2)$
ratsimp(%);
sin(1/(x+1)) = %e*x^2$
b*(a/b-x)+b*x+a;
b*x+b*(a/b-x)+a$
ratsimp(%);
2*a$
((x-1)^(3/2)-(x+1)*sqrt(x-1))/sqrt(x-1)/sqrt(x+1);
((x-1)^(3/2)-sqrt(x-1)*(x+1))/(sqrt(x-1)*sqrt(x+1))$
ratsimp(%);
-2/sqrt(x+1)$
ev(x^(a+1/a),ratsimpexpons);
x^((a^2+1)/a)$
(log(x^2+x)-log(x))^a/log(x+1)^(a/2);
(log(x^2+x)-log(x))^a/log(x+1)^(a/2)$
radcan(%);
log(x+1)^(a/2)$
log(a^(2*x)+2*a^x+1)/log(a^x+1);
log(a^(2*x)+2*a^x+1)/log(a^x+1)$
radcan(%);
2$
(%e^x-1)/(%e^(x/2)+1);
(%e^x-1)/(%e^(x/2)+1)$
radcan(%);
%e^(x/2)-1$
kill(all);
done$
combine(a/x+b/x+a/y+b/y);
(b+a)/y+(b+a)/x$
x/(x-y)^2-1/(x-y)-f(x)/(x-y)^3;
-1/(x-y)+x/(x-y)^2-f(x)/(x-y)^3$
multthru((x-y)^3,%);
-(x-y)^2+x*(x-y)-f(x)$
ratexpand(%);
-y^2+x*y-f(x)$
((a+b)^10*s^2+2*a*b*s+(a*b)^2)/(a*b*s^2);
((b+a)^10*s^2+2*a*b*s+a^2*b^2)/(a*b*s^2)$
multthru(%);
2/s+a*b/s^2+(b+a)^10/(a*b)$
multthru(a . (b+c . (d+e)+f));
a . f+a . c . (e+d)+a . b$
((x+2)^20-2*y)/(x+y)^20+(x+y)^-19-x/(x+y)^20;
1/(y+x)^19+((x+2)^20-2*y)/(y+x)^20-x/(y+x)^20$
xthru(%);
((x+2)^20-y)/(y+x)^20$
2/(x+2)-2/(x+1)+1/(x+1)^2;
2/(x+2)-2/(x+1)+1/(x+1)^2$
ratsimp(%);
-x/(x^3+4*x^2+5*x+2)$
partfrac(%,x);
2/(x+2)-2/(x+1)+1/(x+1)^2$
sort(args(factor(2^63-1)));
''(sort([7^2,73,127,337,92737,649657]))$
factor(z^2*(x+2*y)-4*x-8*y);
(2*y+x)*(z-2)*(z+2)$
x^2*y^2+2*x*y^2+y^2-x^2-2*x-1;
x^2*y^2+2*x*y^2+y^2-x^2-2*x-1$
block([dontfactor:[x]],factor(%/36/(y^2+2*y+1)));
(x^2+2*x+1)*(y-1)/(36*(y+1))$
factor(%e^(3*x)+1);
(%e^x+1)*(%e^(2*x)-%e^x+1)$
factor(x^4+1,a^2-2);
(x^2-a*x+1)*(x^2+a*x+1)$
factor(x^3+x^2*y^2-x*z^2-y^2*z^2);
-((y^2+x)*(z-x)*(z+x))$
(x+2)/(x+3)/(x+b)/(x+c)^2;
(x+2)/((x+3)*(x+b)*(x+c)^2)$
ratsimp(%);
(x+2)/(x^4+(2*c+b+3)*x^3+(c^2+(2*b+6)*c+3*b)*x^2+((b+3)*c^2+6*b*c)*x+3*b*c^2)$
partfrac(%,x);
-((c^2-4*c-b+6)/((c^4+(-2*b-6)*c^3+(b^2+12*b+9)*c^2+(-6*b^2-18*b)*c+9*b^2)
*(x+c)))
-((c-2)/((c^2+(-b-3)*c+3*b)*(x+c)^2))
+(b-2)/(((b-3)*c^2+(6*b-2*b^2)*c+b^3-3*b^2)*(x+b))
-(1/(((b-3)*c^2+(18-6*b)*c+9*b-27)*(x+3)))$
map('factor,%);
-((c^2-4*c-b+6)/((c-3)^2*(c-b)^2*(x+c)))-((c-2)/((c-3)*(c-b)*(x+c)^2))
+(b-2)/((b-3)*(c-b)^2*(x+b))
-(1/((b-3)*(c-3)^2*(x+3)))$
exp:ratsimp((x^5-1)/(x-1));
x^4+x^3+x^2+x+1$
subst(a,x,%);
a^4+a^3+a^2+a+1$
factor(exp,%);
(x-a)*(x-a^2)*(x-a^3)*(x+a^3+a^2+a+1)$
factor(x^12+1);
(x^4+1)*(x^8-x^4+1)$
factor(x^99+1);
(x+1)*(x^2-x+1)*(x^6-x^3+1)*(x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1)
*(x^20+x^19-x^17-x^16+x^14+x^13-x^11-x^10-x^9+x^7+x^6-x^4-x^3+x+1)
*(x^60+x^57-x^51-x^48+x^42+x^39-x^33-x^30-x^27+x^21+x^18-x^12-x^9+x^3+1)$
ev((x+1)*((u+v)^2+a*(w+z)^2),expand);
a*x*z^2+a*z^2+2*a*w*x*z+2*a*w*z+a*w^2*x+v^2*x+2*u*v*x+u^2*x+a*w^2+v^2+2*u*v
+u^2$
factorsum(%);
(x+1)*(a*(z+w)^2+(v+u)^2)$
sqfr(4*x^4+4*x^3-3*x^2-4*x-1);
(2*x+1)^2*(x^2-1)$
gfactor(x^4-1);
(x-1)*(x+1)*(x-%i)*(x+%i)$
factor(y^20-7*y^19+11*y^18-8*y^17+59*y^16-461*y^15+1694*y^14+198*y^13-3541*
y^12+2818*y^11-33327*y^10-11272*y^9-56656*y^8-12672*y^7+433664*y^6+472064*y^5
+241664*y^4+131072*y^3+720896*y^2+1835008*y+1048576);
(y^8-y^7-11*y^6-19*y^5+101*y^4+76*y^3-176*y^2+64*y+256)
*(y^12-6*y^11+16*y^10-39*y^9-19*y^8-75*y^7-315*y^6+300*y^5-304*y^4
+2496*y^3+4096*y^2+6144*y+4096)$
/* sf bug 1650226 */
factor(156*x^7+4808*x^6-182041*x^5-1266489*x^4+43104271*x^3+29839285*x^2-2542327662*x+7826952672);
(x-16)*(x+14)*(x+49)*(2*x-11)*(2*x+21)*(3*x-49)*(13*x-63)$
/* sf bug 681191 */
gfactor(x^4+y^4);
(y^2-%i*x^2)*(y^2+%i*x^2)$
/* sf bug 696804 */
factor(expand((35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19)));
(35*x*y^2+19*x^2+25) * (35*x*y^2+25*x^2+19)$
/* bug reported on mailing list circa 2007-11-27; bignum in exponent */
is (errcatch (rat (x^(2^128))) = []);
false;
/* [ 1943756 ] ezgcd() never returns. */
ezgcd(n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1, -n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1);
[1,n^2+(-2*d+c+b+2)*n+d^2+(-c-b-2)*d+c+b+1,
-n^2+(2*d-c-b-2)*n-d^2+(c+b+2)*d+(-b-1)*c-b-1];
/* [ 1980715 ] wrong radcan-simplification */
radcan(3^(1/6)*9^(1/3));
3^(5/6);
radcan(81^(1/5)*9^(1/3));
3*3^(7/15);
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