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  1.  
  2.  /* This file is to be run by the EXAMPLE command, and may not 
  3.    otherwise work.
  4.    The following are either acceptable lines to maxima, or they are
  5.    two successive '&' characters immediately following a '$' or ';'
  6.    and then followed by the name of the section of examples, and then followed by
  7.    a sequence of maxima forms.
  8. */
  9. &&
  10. ADDITIVE DECLARE(F,ADDITIVE);
  11. F(2*A+3*B);&&
  12. ALGSYS  F1:2*X*(1-L1)-2*(X-1)*L2$
  13. F2:L2-L1$
  14. F3:L1*(1-X**2-Y)$
  15. F4:L2*(Y-(X-1)**2)$
  16. ALGSYS([F1,F2,F3,F4],[X,Y,L1,L2]);
  17. F1:X**2-Y**2$
  18. F2:X**2-X+2*Y**2-Y-1$
  19. ALGSYS([F1,F2],[X,Y]);&&
  20. ALLROOTS  (2*X+1)^3=13.5*(X^5+1);
  21. ALLROOTS(%);&&
  22. ANTISYMMETRIC DECLARE(H,ANTISYMMETRIC);
  23. H(X,Z,Y);&&
  24. APPEND  APPEND([Y+X,0,-3.2],[2.5E20,X]);&&
  25. ARRAYINFO B[1,X]:1$
  26. ARRAY(F,2,3);
  27. ARRAYINFO(B);
  28. ARRAYINFO(F);&&
  29. ARRAYS  A[N]:=N*A[N-1];
  30. A[0]:1$
  31. A[5];
  32. A[N]:=N$
  33. A[6];
  34. A[4];&&
  35. AT  ATVALUE(F(X,Y),[X=0,Y=1],A^2);
  36. ATVALUE('DIFF(F(X,Y),X),X=0,Y+1);
  37. PRINTPROPS(ALL,ATVALUE);
  38. DIFF(4*F(X,Y)^2-U(X,Y)^2,X);
  39. AT(%,[X=0,Y=1]);&&
  40. ATVALUE
  41. kill(f,x,a,u,g);
  42. ATVALUE(F(X,Y),[X=0,Y=1],A^2)$
  43. ATVALUE('DIFF(F(X,Y),X),X=0,Y+1);
  44. PRINTPROPS(ALL,ATVALUE);
  45. DIFF(4*F(X,Y)^2-U(X,Y)^2,X);
  46. AT(%,[X=0,Y=1]);&&
  47. AUGCOEFMATRIX  [2*X-(A-1)*Y=5*B,A*X+B*Y+C=0]$
  48. AUGCOEFMATRIX(%,[X,Y]);&&
  49. BEZOUT  BEZOUT(A*Y+X^2+1,Y^2+X*Y+B,X);
  50. EXPAND(DETERMINANT(%));
  51. %-EXPAND(RESULTANT(A*Y+X^2+1,Y^2+X*Y+B,X));&&
  52. BLOCK
  53.    kill(f);
  54.    HESSIAN(F):=BLOCK([DFXX,DFXY,DFXZ,DFYY,DFYZ,DFZZ],
  55.       DFXX:DIFF(F,X,2),DFXY:DIFF(F,X,1,Y,1),
  56.       DFXZ:DIFF(F,X,1,Z,1),DFYY:DIFF(F,Y,2),
  57.       DFYZ:DIFF(F,Y,1,Z,1),DFZZ:DIFF(F,Z,2),
  58.       DETERMINANT(MATRIX([DFXX,DFXY,DFXZ],[DFXY,DFYY,DFYZ],
  59.               [DFXZ,DFYZ,DFZZ])))$
  60. kill(x,y,z);
  61. HESSIAN(X^3-3*A*X*Y*Z+Y^3);
  62. SUBST(1,Z,QUOTIENT(%,-54*A^2));
  63. F(X):=BLOCK([Y,use_fast_arrays:FALSE], LOCAL(A), Y:4, A[Y]:X, DISPLAY(A[Y]))$
  64. Y:2$
  65. A[Y+2]:0$
  66. F(9);
  67. Ff(X):=BLOCK([Y,A,use_fast_arrays:true], Y:4, A[Y]:X, DISPLAY(A[Y]))$
  68. Ff(10);
  69. A[Y+2];&&
  70. BOTHCOEFF  ISLINEAR(EXP,VAR):=BLOCK([C],
  71.             C:BOTHCOEF(RAT(EXP,VAR),VAR),
  72.             IS(FREEOF(VAR,C) AND C[1]#0))$
  73. ISLINEAR((R^2-(X-R)^2)/X,X);&&
  74. CATCH  G(L):=CATCH(MAP(LAMBDA([X],IF X<0 THEN THROW(X) ELSE F(X)),L))$
  75. G([1,2,3,7]);
  76. G([1,2,-3,7]);&&
  77. CF  CF([1,2,-3]+[1,-2,1]);
  78. CFDISREP(%);
  79. CFLENGTH:4$
  80. CF(SQRT(3));
  81. CFEXPAND(%);
  82. EV(%[1,2]/%[2,2],NUMER);&&
  83. CFDISREP  CF([1,2,-3]+[1,-2,1]);
  84. CFDISREP(%);&&
  85. CFEXPAND  CFLENGTH:4$
  86. CF(SQRT(3));
  87. CFEXPAND(%);
  88. EV(%[1,2]/%[2,2],NUMER);&&
  89. CHANGEVAR  'INTEGRATE(%E^(SQRT(A)*SQRT(Y)),Y,0,4);
  90. CHANGEVAR(%,Y-Z^2/A,Z,Y);&&
  91. CHARPOLY  A:MATRIX([3,1],[2,4]);
  92. EXPAND(CHARPOLY(A,LAMBDA));
  93. (PROGRAMMODE:TRUE,SOLVE(%));
  94. MATRIX([X1],[X2]);
  95. EV(A.%-LAMBDA*%,%TH(2)[1]);
  96. %[1,1]=0;
  97. X1^2+X2^2=1;
  98. SOLVE([%TH(2),%],[X1,X2]);&&
  99. COEFF  COEFF(2*A*TAN(X)+TAN(X)+B=5*TAN(X)+3,TAN(X));
  100. COEFF(Y+X*%E^X+1,X,0);&&
  101. COMBINE  COMBINE(A/X+B/X+A/Y+B/Y);&&
  102. COMMUTATIVE DECLARE(H,COMMUTATIVE);
  103. H(X,Z,Y);&&
  104. COMPLEX  (SQRT(-4)+SQRT(2.25))^2;
  105. EXPAND(%);
  106. EXPAND(SQRT(2*%I));&&
  107. CONTENT  CONTENT(2*X*Y+4*X^2*Y^2,Y);&&
  108. DEFMATCH  NONZEROANDFREEOF(X,E):=IS(E#0 AND FREEOF(X,E));
  109. MATCHDECLARE(A,NONZEROANDFREEOF(X),B,FREEOF(X));
  110. DEFMATCH(LINEAR,A*X+B,X);
  111. LINEAR(3*Z+(Y+1)*Z+Y**2,Z);
  112. MATCHDECLARE([A,F],TRUE);
  113. CONSTINTERVAL(L,H):=CONSTANTP(H-L)$
  114. MATCHDECLARE(B,CONSTINTERVAL(A))$
  115. MATCHDECLARE(X,ATOM)$
  116. BLOCK(REMOVE(INTEGRATE,OUTATIVE),
  117.       DEFMATCH(CHECKLIMITS,'INTEGRATE(F,X,A,B)),
  118.       DECLARE(INTEGRATE,OUTATIVE))$
  119. 'INTEGRATE(SIN(T),T,X+%PI,X+2*%PI)$
  120. CHECKLIMITS(%);
  121. 'INTEGRATE(SIN(T),T,0,X)$
  122. CHECKLIMITS(%);
  123. REMVALUE(A,B,F,X)$&&
  124. DEFTAYLOR  DEFTAYLOR(F(X),X^2+SUM(X^I/(2^I*I!^2),I,4,INF));
  125. TAYLOR(%E^SQRT(F(X)),X,0,4);&&
  126. DELETE  DELETE(SIN(X),X+SIN(X)+Y);&&
  127. DEPENDS  
  128. kill(a,x,f,y,t);
  129. DEPENDS(A,X);
  130. DIFF(A.A,X);
  131. DEPENDS(F,[X,Y],[X,Y],T);
  132. DIFF(F,T);&&
  133. DERIVDEGREE  'DIFF(Y,X,2)+'DIFF(Y,Z,3)*2+'DIFF(Y,X)*X^2;
  134. DERIVDEGREE(%,Y,X);&&
  135. DESOLVE  EQN1:'DIFF(F(X),X)='DIFF(G(X),X)+SIN(X);
  136. EQN2:'DIFF(G(X),X,2)='DIFF(F(X),X)-COS(X);
  137. ATVALUE('DIFF(G(X),X),X=0,A);
  138. ATVALUE(F(X),X=0,1);
  139. DESOLVE([EQN1,EQN2],[F(X),G(X)]);
  140. /* VERIFICATION */
  141. [EQN1,EQN2],%,DIFF;&&
  142. DIFF  kill(f,g,h,x,y);
  143. DIFF(SIN(X)+X^3+2*X^2,X);
  144. DIFF(SIN(X)*COS(X),X);
  145. DIFF(SIN(X)*COS(X),X,2);
  146. DERIVABBREV:TRUE$
  147. DIFF(EXP(F(X)),X,2);
  148. 'INTEGRATE(F(X,Y),Y,G(X),H(X));
  149. DIFF(%,X);&&
  150. DISPLAY  DISPLAY(B[1,2]);&&
  151. DIVIDE  DIVIDE(X+Y,X-Y,X);
  152. DIVIDE(X+Y,X-Y);&&
  153. DO  FOR A:-3 THRU 26 STEP 7 DO LDISPLAY(A)$
  154. S:0$
  155. FOR I:1 WHILE I<=10 DO S:S+I;
  156. S;
  157. SERIES:1$
  158. TERM:EXP(SIN(X))$
  159. FOR P:1 UNLESS P>7 DO
  160.           (TERM:DIFF(TERM,X)/P,
  161.           SERIES:SERIES+SUBST(X=0,TERM)*X^P)$
  162. SERIES;
  163. POLY:0$
  164. FOR I:1 THRU 5 DO
  165.         FOR J:I STEP -1 THRU 1 DO
  166.            POLY:POLY+I*X^J$
  167. POLY;
  168. GUESS:-3.0$
  169. FOR I THRU 10 DO (GUESS:SUBST(GUESS,X,0.5*(X+10/X)),
  170.          IF ABS(GUESS^2-10)<0.00005 THEN RETURN(GUESS));
  171. FOR COUNT:2 NEXT 3*COUNT THRU 20
  172.          DO LDISPLAY(COUNT)$
  173. X:1000;
  174. THRU 10 WHILE X#0 DO X:0.5*(X+5/X)$
  175. X;
  176. REMVALUE(X);
  177. NEWTON(F,GUESS):=BLOCK([NUMER,Y],local(f,df,x,guess),
  178.      NUMER:TRUE,
  179.         DEFINE(DF(X),DIFF(F(X),X)),
  180.     DO (Y:DF(GUESS), IF Y=0 THEN ERROR(
  181.           "derivative at",GUESS,"is zero"),
  182.          GUESS:GUESS-F(GUESS)/Y,
  183.          IF ABS(F(GUESS))<5.0E-6 THEN RETURN(GUESS)))$
  184. SQR(X):=X^2-5.0$
  185. NEWTON(SQR,1000);
  186. FOR F IN [LOG, RHO, ATAN] DO LDISP(F(1.0))$
  187. EV(CONCAT(E,LINENUM-1),NUMER);&&
  188. DOTSCRULES  DECLARE(L,SCALAR,[M1,M2,M3],NONSCALAR);
  189. EXPAND((1-L*M1).(1-L*M2).(1-L*M3));
  190. %,DOTSCRULES;
  191. RAT(%,L);&&
  192. DPART  DPART(X+Y/Z^2,1,2,1);
  193. EXPAND((B+A)^4);
  194. (B+A)^2*(Y+X)^2;
  195. EXPAND(%);
  196. %TH(3)/%;
  197. FACTOR(%);
  198. DPART(%TH(2),2,4);
  199. PART(%TH(3),2,4);&&
  200. ECHELON  MATRIX([2,1-A,-5*B],[A,B,C]);
  201. ECHELON(%);&&
  202. ELIMINATE  EXP1:2*X^2+Y*X+Z;
  203. EXP2:3*X+5*Y-Z-1;
  204. EXP3:Z^2+X-Y^2+5;
  205. ELIMINATE([EXP3,EXP2,EXP1],[Y,Z]);&&
  206. ENTERMATRIX  ENTERMATRIX(2,1);&&
  207. EQUATIONS  X+1=Y^2;
  208. X-1=2*Y+1$
  209. %TH(2)+%;
  210. %TH(3)/Y;
  211. 1/%;&&
  212. EV
  213. kill(y,x,w);
  214. SIN(X)+COS(Y)+(W+1)^2+'DIFF(SIN(W),W);
  215. EV(%,SIN,EXPAND,DIFF,X=2,Y=1);
  216. EV(X+Y,X:A+Y,Y:2);
  217. 'DIFF(Y^2+X*Y+X^2,X,2,Y,1);
  218.  
  219. EV(%,DIFF);
  220. 2*X-3*Y=3$
  221. -3*X+2*Y=-4$
  222. SOLVE([%TH(2),%]);
  223. EV(%TH(3),%);
  224. X+1/X>GAMMA(1/2);
  225. EV(%,NUMER,X=1/2);
  226. EV(%,PRED);&&
  227. EVALUATION  DIFF(X*F(X),X);
  228. F(X):=SIN(X)$
  229. EV(%TH(2),DIFF);
  230. X;
  231. X:3$
  232. X;
  233. 'X;
  234. F(X):=X^2;
  235. 'F(2);
  236. EV(%,F);
  237. '(F(2));
  238. ''%;
  239. SUM(I!,I,1,4);
  240. 'SUM(I!,I,1,4);
  241. REMVALUE(X);
  242. 'INTEGRATE(F(X),X,A,B);
  243. FOR I THRU 5 DO S:S+I^2;
  244. S;
  245. EV(%,S:0);
  246. EV(%TH(2));
  247. 'SUM(G(I),I,0,N);
  248. Z*%E^Z;
  249. EV(%,Z:X^2);
  250. SUBST(X^2,Z,%TH(3));
  251. A:%;
  252. A+1;
  253. KILL(A,y);
  254. A;
  255. /* DECLARE(INTEGRATE,NOUN)$ */
  256. INTEGRATE(Y^2,Y);
  257. ''INTEGRATE(Y^2,Y);
  258. F(Y):=DIFF(Y*LOG(Y),Y,2);
  259. F(Y):=''(DIFF(Y*LOG(Y),Y,2));
  260. ''(CONCAT(C,LINENUM-1));
  261. (X+Y)^3$
  262. DIFF(%,X);
  263. Y:X^2+1$
  264. ''(CONCAT(C,LINENUM-2));&&
  265. EVENFUN DECLARE(G,EVENFUN);
  266. G(-X);&&
  267. EXP  EV(%E^X*SIN(X)^2,EXPONENTIALIZE);
  268. kill(x);
  269. INTEGRATE(%,X);
  270. EV(%,DEMOIVRE);
  271. ANS:EV(%,RATEXPAND);
  272. EV(%,X:1,NUMER)-EV(%,X:0,NUMER);
  273. INTEGRATE(%E^X*SIN(X)^2,X);
  274. TRIGREDUCE(%);
  275. %-ANS;
  276. EV(SIN(X),%EMODE);&&
  277. EXPAND  (1/(X+Y)^4-3/(Y+Z)^3)^2;
  278. EXPAND(%,2,0);
  279. EXPAND(A.(B+C.(D+E)+F));
  280. EXPAND((X+1)^3);
  281. (X+1)^7;
  282. EXPAND(%);
  283. EXPAND(%TH(2),7,7);
  284. EV(A*(B+C)+A*(B+C)^2,EXPOP:1);&&
  285. FACTCOMB  (N+1)^2*N!^2;
  286. FACTCOMB(%);&&
  287. FACTOR  FACTOR(2^63-1);
  288. FACTOR(Z^2*(X+2*Y)-4*X-8*Y);
  289. X^2*Y^2+2*X*Y^2+Y^2-X^2-2*X-1;
  290. BLOCK([DONTFACTOR:[X]],FACTOR(%/36/(Y^2+2*Y+1)));
  291. FACTOR(%E^(3*X)+1);
  292. FACTOR(X^4+1,A^2-2);
  293. FACTOR(X^3+X^2*Y^2-X*Z^2-Y^2*Z^2);
  294. (X+2)/(X+3)/(X+B)/(X+C)^2;
  295. RATSIMP(%);
  296. PARTFRAC(%,X);
  297. MAP('FACTOR,%);
  298. RATSIMP((X^5-1)/(X-1));
  299. SUBST(A,X,%);
  300. FACTOR(%TH(2),%);
  301. FACTOR(X^12+1);
  302. FACTOR(X^99+1);&&
  303. FACTORSUM  EV((X+1)*((U+V)^2+A*(W+Z)^2),EXPAND);
  304. FACTORSUM(%);&&
  305. FEATUREP DECLARE(J,EVEN)$
  306. FEATUREP(J,INTEGER);&&
  307. FREEOF  FREEOF(Y,SIN(X+2*Y));
  308. FREEOF(COS(Y),"*",SIN(Y)+COS(X));&&
  309. FULLMAP  FULLMAP(G,A+B*C);
  310. MAP(G,A+B*C);&&
  311. FULLMAPL  FULLMAPL("+",[3,[4,5]],[[A,1],[0,-1.5]]);&&
  312. FUNCSOLVE  FUNCSOLVE((N+1)*F(N)-(N+3)*F(N+1)/(N+1)=(N-1)/(N+2),F(N));&&
  313. functions
  314. KILL(X,Y,F,G,H);
  315. F(X):=X^2+Y;
  316. F(2);
  317. EV(F(2),Y:7);
  318. F(X):=SIN(X)^2+1;
  319. F(X+1);
  320. G(Y,Z):=F(Z)+3*Y;
  321. EV(G(2*Y+Z,-0.5),Y:7);
  322. H(N):=SUM(I*X^I,I,0,N);
  323. FUNCTIONS;
  324. T[N](X):=RATEXPAND(2*X*T[N-1](X)-T[N-2](X));
  325. T[0](X):=1$
  326. T[1](X):=X$
  327. T[4](Y);
  328. G[N](X):=SUM(EV(X),I,N,N+2);
  329. H(N,X):=SUM(EV(X),I,N,N+2);
  330. G[2](I^2);
  331. H(2,I^2);
  332. P[N](X):=RATSIMP(1/(2^N*N!)*DIFF((X^2-1)^N,X,N));
  333. Q(N,X):=RATSIMP(1/(2^N*N!)*DIFF((X^2-1)^N,X,N));
  334. P[2];
  335. P[2](Y+1);
  336. Q(2,Y);
  337. P[2](5);
  338. F[I,J](X,Y):=X^I+Y^J;
  339. G(FUN,A,B):=PRINT(FUN," applied to ",A," and ",B," is ",FUN(A,B))$
  340. G(F[2,1],SIN(%PI),2*C);&&
  341. GENMATRIX  H[I,J]:=1/(I+J-1)$
  342. GENMATRIX(H,3,3);&&
  343. GET  PUT(%E,TRANSCENDENTAL,TYPE);
  344. PUT(%PI,TRANSCENDENTAL,TYPE)$
  345. PUT(%I,ALGEBRAIC,TYPE)$
  346. TYPEOF(X):=BLOCK([Q], IF NUMBERP(X)
  347.                THEN RETURN(ALGEBRAIC),
  348.                IF NOT ATOM(X)
  349.                THEN RETURN(MAPLIST(TYPEOF,X)),
  350.                Q:GET(X,TYPE), IF Q=FALSE THEN
  351.                ERROR("NOT NUMERIC") ELSE Q)$
  352. ERRCATCH(TYPEOF(2*%E+X*%PI));
  353. TYPEOF(2*%E+%PI);&&
  354. GFACTOR  GFACTOR(X^4-1);&&
  355. GRADEF  DEPENDS(Y,X);
  356. kill(f,g,j);
  357. GRADEF(F(X,Y),X^2,G(X,Y));
  358. DIFF(F(X,Y),X);
  359. GRADEF(J(N,Z),'DIFF(J(N,Z),N),
  360.     J(N-1,Z)-N/Z*J(N,Z))$
  361. RATSIMP(DIFF(J(2,X),X,2));&&
  362. HORNER  POLY:1.0E-20*X^2-5.5*X+5.2E20;
  363. ERRCATCH(EV(%,X=1.0E20));
  364. HORNER(POLY,X),KEEPFLOAT;
  365. EV(%,X=1.0E20);&&
  366. IF  FIB[N]:=IF N=1 OR N=2 THEN 1 ELSE FIB[N-1]+FIB[N-2];
  367. FIB[1]+FIB[2];
  368. FIB[3];
  369. FIB[5];
  370. ETA(MU,NU):=IF MU=NU THEN MU ELSE IF MU>NU THEN MU-NU ELSE MU+NU;
  371. ETA(5,6);
  372. ETA(ETA(7,7),ETA(1,2));
  373. IF NOT 5>=2 AND 6<=5 OR 4+1>3 THEN A ELSE B;&&
  374. ILT  'INTEGRATE(SINH(A*X)*F(T-X),X,0,T)+B*F(T)=T^2;
  375. LAPLACE(%,T,S);
  376. LINSOLVE([%],['LAPLACE(F(T),T,S)]);
  377. ILT(EV(%[1]),S,T);&&
  378. INPART  X+Y+W*Z;
  379. INPART(%,3,2);
  380. 'LIMIT(F(X)^G(X+1),X,0,MINUS);
  381. INPART(%,1,2);&&
  382. INTEGRATE
  383. test(f):=block([u],u:integrate(f,x),ratsimp(f-diff(u,x)));
  384. test(sin(x));
  385. test(1/(1+x));
  386. test(1/(1+x^2));
  387. INTEGRATE(SIN(X)^3,X);
  388. kill(q)$
  389. INTEGRATE(%E^X/(%E^X+2),X);
  390. INTEGRATE(1/(X*LOG(X)),X);
  391. INTEGRATE(SIN(2*X+3),X);
  392. INTEGRATE(%E^X*ERF(X),X);
  393. INTEGRATE(X/(X^3+1),X);
  394. DIFF(%,X);
  395. RATSIMP(%);
  396. INTEGRATE(X^(5/4)/(X+1)^(5/2),X,0,INF);
  397. GRADEF(Q(X),SIN(X^2));
  398. DIFF(LOG(Q(R(X))),X);
  399. INTEGRATE(%,X);&&
  400. IS  IS(X^2>=2*X-1);
  401. ASSUME(A>1);
  402. IS(LOG(LOG(A+1)+1)>0 AND A^2+1>2*A);&&
  403. ISOLATE  (A+B)^4*(1+X*(2*X+(C+D)^2));
  404. ISOLATE(%,X);
  405. RATEXPAND(%)$
  406. EV(%);
  407. (A+B)*(X+A+B)^2*%E^(X^2+A*X+B);
  408. ISOLATE(%,X),EXPTISOLATE:TRUE;&&
  409. LAMBDA  LAMBDA([X,Y,Z],X^2+Y^2+Z^2);
  410. %(1,2,A);
  411. "+"(1,2,A);&&
  412. LAPLACE  LAPLACE(%E^(2*T+A)*SIN(T)*T,T,S);&&
  413. LASSOCIATIVE DECLARE(G,LASSOCIATIVE);
  414. G(G(A,B),G(C,D));
  415. G(G(A,B),G(C,D))-G(A,G(B,G(C,D)));&&
  416. LET  MATCHDECLARE([A,A1,A2],TRUE);
  417. ONELESS(X,Y):=IS(X=Y-1)$
  418. LET(A1*A2!,A1!,ONELESS,A2,A1);
  419. LET(A1!/A1,(A1-1)!),LETRAT;
  420. LETSIMP(N*M!*(N-1)!/M),LETRAT;
  421. LET(SIN(A)^2,1-COS(A)^2);
  422. SIN(X)^4;
  423. LETSIMP(%);&&
  424. LETRULES  MATCHDECLARE([A,A1,A2],TRUE);
  425. ONELESS(X,Y):=IS(X=Y-1)$
  426. LET(A1*A2!,A1!,ONELESS,A2,A1);
  427. LET(A1!/A1,(A1-1)!),LETRAT;
  428. LETSIMP(N*M!*(N-1)!/M),LETRAT;
  429. LET(SIN(A)^2,1-COS(A)^2);
  430. SIN(X)^4;
  431. LETSIMP(%);&&
  432. LIMIT  LIMIT(X*LOG(X),X,0,PLUS);
  433. LIMIT((1+X)^(1/X),X,0);
  434. LIMIT(%E^X/X,X,INF);
  435. LIMIT(SIN(1/X),X,0);&&
  436. LINEAR DECLARE(F,LINEAR);
  437. F(2*A+3*B);
  438. F(2*X+Y,X);&&
  439. LINSOLVE  X+Z=Y$
  440. 2*A*X-Y=2*A^2$
  441. Y-2*Z=2$
  442. LINSOLVE([%TH(3),%TH(2),%],[X,Y,Z]),GLOBALSOLVE;&&
  443. LISTOFVARS  LISTOFVARS(F(X[1]+Y)/G^(2+A));&&
  444. LISTS  [X^2,Y/3,-2];
  445. %[1]*X;
  446. [A,%TH(2),%];&&
  447. LOGCONTRACT  2*(A*LOG(X) + 2*A*LOG(Y));
  448. LOGCONTRACT(%);
  449. LOGCONTRACT(LOG(SQRT(X+1)+SQRT(X)) + LOG(SQRT(X+1)-SQRT(X)));&&
  450. MAP  MAP(F,X+A*Y+B*Z);
  451. MAP(LAMBDA([U],PARTFRAC(U,X)),X+1/(X^3+4*X^2+5*X+2));
  452. MAP(RATSIMP, X/(X^2+X)+(Y^2+Y)/Y);
  453. MAP("=",[A,B],[-0.5,3]);&&
  454. MATCHDECLARE  MATCHDECLARE(A,TRUE)$
  455. TELLSIMP(SIN(A)^2,1-COS(A)^2)$
  456. SIN(Y)^2;
  457. KILL(RULES);
  458. NONZEROANDFREEOF(X,E):=IS(E#0 AND FREEOF(X,E));
  459. MATCHDECLARE(A,NONZEROANDFREEOF(X),B,FREEOF(X));
  460. DEFMATCH(LINEAR,A*X+B,X);
  461. LINEAR(3*Z+(Y+1)*Z+Y**2,Z);
  462. MATCHDECLARE([A,F],TRUE);
  463. CONSTINTERVAL(L,H):=CONSTANTP(H-L)$
  464. MATCHDECLARE(B,CONSTINTERVAL(A))$
  465. MATCHDECLARE(X,ATOM)$
  466. BLOCK(REMOVE(INTEGRATE,OUTATIVE),
  467.       DEFMATCH(CHECKLIMITS,'INTEGRATE(F,X,A,B)),
  468.       DECLARE(INTEGRATE,OUTATIVE))$
  469. 'INTEGRATE(SIN(T),T,X+%PI,X+2*%PI)$
  470. CHECKLIMITS(%);
  471. 'INTEGRATE(SIN(T),T,0,X)$
  472. CHECKLIMITS(%);&&
  473. MATRICES  M:MATRIX([A,0],[B,1]);
  474. M^2;
  475. M.M;
  476. M[1,1]*M;
  477. %-%TH(2)+1;
  478. M^^-1;
  479. [X,Y].M;
  480. MATRIX([A,B,C],[D,E,F],[G,H,I]);
  481. %^^2;&&
  482. MINFACTORIAL  N!/(N+1)!;
  483. MINFACTORIAL(%);&&
  484. MULTIPLICATIVE DECLARE(F,MULTIPLICATIVE);
  485. F(2*A*B);&&
  486. MULTTHRU  X/(X-Y)^2-1/(X-Y)-F(X)/(X-Y)^3;
  487. MULTTHRU((X-Y)^3,%);
  488. RATEXPAND(%);
  489. ((A+B)^10*S^2+2*A*B*S+(A*B)^2)/(A*B*S^2);
  490. MULTTHRU(%);
  491. MULTTHRU(A.(B+C.(D+E)+F));&&
  492. NARY DECLARE(J,NARY);
  493. J(J(A,B),J(C,D));&&
  494. NOUNIFY  'LIMIT(F(X)^G(X+1),X,0,MINUS);
  495. IS(INPART(%,0)=NOUNIFY(LIMIT));&&
  496. NROOTS  X^10-2*X^4+1/2;
  497. NROOTS(%,-6,9.1);&&
  498. NUMFACTOR  GAMMA(7/2);
  499. NUMFACTOR(%);&&
  500. NUSUM  NUSUM(N*N!,N,0,N);
  501. NUSUM(N^4*4^N/BINOMIAL(2*N,N),N,0,N);
  502. UNSUM(%,N);
  503. UNSUM(PROD(I^2,I,1,N),N);
  504. NUSUM(%,N,1,N);&&
  505. ODDFUN DECLARE(F,ODDFUN);
  506. F(-X);&&
  507. ODE2  X^2*'DIFF(Y,X) + 3*X*Y = SIN(X)/X;
  508. SOLN1:ODE2(%,Y,X);
  509. IC1(SOLN1,X=%PI,Y=0);
  510. 'DIFF(Y,X,2) + Y*'DIFF(Y,X)^3 = 0;
  511. SOLN2:ODE2(%,Y,X);
  512. RATSIMP(IC2(SOLN2,X=0,Y=0,'DIFF(Y,X)=2));
  513. BC2(SOLN2,X=0,Y=1,X=1,Y=3);&&
  514. OPTIMIZE  DIFF(EXP(X^2+Y)/(X+Y),X,2);
  515. OPTIMIZE(%);&&
  516. ORDERGREAT  A^2+B*X;
  517. ORDERGREAT(A);
  518. A^2+B*X;
  519. %-%TH(3);
  520. UNORDER();&&
  521. ORDERLESS  Y^2+B*X;
  522. ORDERLESS(Y);
  523. Y^2+B*X;
  524. %-%TH(3);
  525. UNORDER();&&
  526. OUTATIVE DECLARE(F,OUTATIVE);
  527. F(2*A);&&
  528. PART  X+Y/Z^2;
  529. PART(%,1,2,2);
  530. REMVALUE(X);
  531. 'INTEGRATE(F(X),X,A,B)+X;
  532. PART(%,1,1);
  533. X^2+2*X=Y^2;
  534. %+1;
  535. LHS(%);
  536. PART(%TH(2),2);
  537. PART(%,1);
  538. 27*Y^3+54*X*Y^2+36*X^2*Y+Y+8*X^3+X+1;
  539. PART(%,2,[1,3]);
  540. SQRT(PIECE/54);&&
  541. PARTFRAC  2/(X+2)-2/(X+1)+1/(X+1)^2;
  542. RATSIMP(%);
  543. PARTFRAC(%,X);&&
  544. PARTITION  PARTITION(2*A*X*F(X),X);
  545. PARTITION(A+B,X);&&
  546. PICKAPART  INTEGRATE(1/(X^3+2),X)$
  547. PICKAPART(%,1);&&
  548. POISSIMP  PFEFORMAT:TRUE$
  549. POISSIMP(SIN(X)^2);
  550. (2*A^2-B)*COS(X+2*Y)-(A*B+5)*SIN(U-4*X);
  551. POISEXPT(%,2)$
  552. PRINTPOIS(%);
  553. POISINT(%TH(2),Y)$
  554. POISSIMP(%);
  555. POISSIMP(SIN(X)^5+COS(X)^5);
  556. PFEFORMAT:FALSE$&&
  557. POLARFORM  RECTFORM(SIN(2*%I+X));
  558. POLARFORM(%);
  559. RECTFORM(LOG(3+4*%I));
  560. POLARFORM(%);
  561. RECTFORM((2+3.5*%I)^0.25),NUMER;
  562. POLARFORM(%);&&
  563. POLY_discriminant  FACTOR(POLY_DISCRIMINANT((X-A)*(X-B)*(X-C),X));&&
  564. POSFUN DECLARE(F,POSFUN);
  565. IS(F(X)>0);&&
  566. POWERSERIES  POWERSERIES(LOG(SIN(X)/X),X,0);&&
  567. PRINTPROPS GRADEF(R,X,X/R)$
  568. GRADEF(R,Y,Y/R)$
  569. PRINTPROPS(R,ATOMGRAD);
  570. PROPVARS(ATOMGRAD);&&
  571. PRODUCT  PRODUCT(X+I*(I+1)/2,I,1,4);&&
  572. PROPERTIES PROPERTIES(CONS);
  573. ASSUME(VAR1>0);
  574. PROPERTIES(VAR1);
  575. VAR2:2$
  576. PROPERTIES(VAR2);&&
  577. PROPVARS GRADEF(R,X,X/R)$
  578. GRADEF(R,Y,Y/R)$
  579. PRINTPROPS(R,ATOMGRAD);
  580. PROPVARS(ATOMGRAD);&&
  581. QUNIT  QUNIT(17);
  582. EXPAND(%*(SQRT(17)-4));&&
  583. RADCAN  (LOG(X^2+X)-LOG(X))^A/LOG(X+1)^(A/2);
  584. RADCAN(%);
  585. LOG(A^(2*X)+2*A^X+1)/LOG(A^X+1);
  586. RADCAN(%);
  587. (%E^X-1)/(%E^(X/2)+1);
  588. RADCAN(%);&&
  589. RANK  MATRIX([2,1-A,-5*B],[A,B,C]);
  590. RANK(%);&&
  591. RASSOCIATIVE DECLARE(G,RASSOCIATIVE);
  592. G(G(A,B),G(C,D));
  593. G(G(A,B),G(C,D))-G(A,G(B,G(C,D)));&&
  594. RAT  RAT(X^2);
  595. DIFF(F(%),X);
  596. ((X-2*Y)^4/(X^2-4*Y^2)^2+1)*(Y+A)*(2*Y+X)/(4*Y^2+X^2);
  597. RAT(%,Y,A,X);
  598. (X+3)^20;
  599. RAT(%);
  600. DIFF(%,X);
  601. FACTOR(%);&&
  602. RATCOEFF  A*X+B*X+5$
  603. RATCOEF(%,A+B);&&
  604. RATDIFF  (4*X^3+10*X-11)/(X^5+5);
  605. MOD(%),MODULUS:3;
  606. RATDIFF(%TH(2),X);&&
  607. RATEXPAND  RATEXPAND((2*X-3*Y)^3);
  608. (X-1)/(X+1)^2+1/(X-1);
  609. EXPAND(%);
  610. RATEXPAND(%TH(2));&&
  611. RATSIMP  SIN(X/(X^2+X))=%E^((LOG(X)+1)^2-LOG(X)^2);
  612. RATSIMP(%);
  613. B*(A/B-X)+B*X+A;
  614. RATSIMP(%);
  615. ((X-1)^(3/2)-(X+1)*SQRT(X-1))/SQRT(X-1)/SQRT(X+1);
  616. RATSIMP(%);
  617. X^(A+1/A),RATSIMPEXPONS;&&
  618. RATSUBST  RATSUBST(A,X*Y^2,X^4*Y^8+X^4*Y^3);
  619. 1 + COS(X) + COS(X)^2 + COS(X)^3 + COS(X)^4;
  620. RATSUBST(1-SIN(X)^2,COS(X)^2,%);
  621. RATSUBST(1-COS(X)^2,SIN(X)^2,SIN(X)^4);&&
  622. RATWEIGHT  RATWEIGHT(A,1,B,1);
  623. RAT(A+B+1);
  624. %^2;
  625. EV(%TH(2)^2,RATWTLVL:1);&&
  626. REALPART  (%I*V+U)/(F+%I*E)+%E^(%I*ALPHA);
  627. REALPART(%);&&
  628. REALROOTS  REALROOTS(X^5-X-1,5.0E-6);
  629. %[1],FLOAT;
  630. X^5-X-1,%;&&
  631. RESIDUE  RESIDUE(S/(S^2+A^2),S,A*%I);
  632. RESIDUE(SIN(A*X)/X^4,X,0);&&
  633. RESULTANT  RESULTANT(A*Y+X^2+1,Y^2+X*Y+B,X);&&
  634. REVEAL  INTEGRATE(1/(X^3+2),X)$
  635. REVEAL(%,2);
  636. REVEAL(%TH(2),3);&&
  637. REVERSE  UNION(X,Y):=IF X=[] THEN Y ELSE
  638.                IF MEMBER(T:FIRST(X),Y) THEN UNION(REST(X),Y)
  639.                ELSE CONS(T,UNION(REST(X),Y))$
  640. UNION([A,B,1,1/2,X^2],[-X^2,A,Y,1/2]);
  641. BERNPOLY(X,5);
  642. MAPLIST(NUMFACTOR,%);
  643. APPLY(MIN,%);&&
  644. RISCH  
  645. RISCH(X^2*ERF(X),X);
  646. DIFF(%,X),RATSIMP;&&
  647. ROOTSCONTRACT ROOTSCONMODE:FALSE$
  648. ROOTSCONTRACT(X^(1/2)*Y^(3/2));
  649. ROOTSCONTRACT(X^(1/2)*Y^(1/4));
  650. ROOTSCONMODE:TRUE$
  651. ROOTSCONTRACT(X^(1/2)*Y^(1/4));
  652. ROOTSCONTRACT(X^(1/2)*Y^(1/3));
  653. ROOTSCONMODE:ALL$
  654. ROOTSCONTRACT(X^(1/2)*Y^(1/4));
  655. ROOTSCONTRACT(X^(1/2)*Y^(1/3));
  656. ROOTSCONMODE:FALSE$
  657. ROOTSCONTRACT(SQRT(SQRT(X+1)+SQRT(X))*SQRT(SQRT(X+1)-SQRT(X))); 
  658. ROOTSCONMODE:TRUE$
  659. ROOTSCONTRACT(SQRT(SQRT(5)+5)-5^(1/4)*SQRT(SQRT(5)+1));&&
  660. SCANMAP  (A^2+2*A+1)*Y+X^2;
  661. SCANMAP(FACTOR,%);
  662. SCANMAP(FACTOR,EXPAND(%TH(2)));
  663. U*V^(A*X+B)+C;
  664. SCANMAP('F,%);&&
  665. SCSIMP  EXP:K^2*N^2+K^2*M^2*N^2-K^2*L^2*N^2-K^2*L^2*M^2*N^2;
  666. EQ1:K^2+L^2=1;
  667. EQ2:N^2-M^2=1;
  668. SCSIMP(EXP,EQ1,EQ2);
  669. EXQ:(K1*K4-K1*K2-K2*K3)/K3^2;
  670. EQ3:K1*K4-K2*K3=0;
  671. EQ4:K1*K2+K3*K4=0;
  672. SCSIMP(EXQ,EQ3,EQ4);&&
  673. SOLVE  SOLVE(ASIN(COS(3*X))*(F(X)-1),X);
  674. SOLVE(5^F(X)=125,F(X)),SOLVERADCAN;
  675. [4*X^2-Y^2=12,X*Y-X=2];
  676. SOLVE(%,[X,Y]);
  677. SOLVE(X^3+A*X+1,X);
  678. SOLVE(X^3-1);
  679. SOLVE(X^6-1);
  680. EV(X^6-1,%[1]);
  681. EXPAND(%);
  682. X^2-1;
  683. SOLVE(%,X);
  684. %TH(2),%[1];&&
  685. SPECINT  ASSUME(P>0,A>0)$
  686. /* a Laplace transform */
  687. T^(1/2)*%E^(-A*T/4)*%E^(-P*T);
  688. SPECINT(%,T);
  689. /* a Bessel function */
  690. T^(1/2)*%J[1](2*A^(1/2)*T^(1/2))*%E^(-P*T);
  691. SPECINT(%,T);
  692. FORGET(P>0,A>0)$&&
  693. SQFR  SQFR(4*X^4+4*X^3-3*X^2-4*X-1);&&
  694. SUBSTINPART  X.'DIFF(F(X),X,2);
  695. SUBSTINPART(D^2,%,2);
  696. SUBSTINPART(F1,F[1](X+1),0);&&
  697. SUBSTITUTE  SUBST(A,X+Y,X+(X+Y)^2+Y);
  698. SUBST(-%I,%I,A+B*%I);
  699. SUBST(X,Y,X+Y);
  700. SUBST(X=0,DIFF(SIN(X),X));
  701. ERRCATCH(EV(DIFF(SIN(X),X),X=0));
  702. INTEGRATE(X^I,X),I=-1;
  703. ERRCATCH(SUBST(-1,I,INTEGRATE(X^I,X)));
  704. MATRIX([A,B],[C,D]);
  705. SUBST("[",MATRIX,%);&&
  706. SUBSTPART  1/(X^2+2);
  707. SUBSTPART(3/2,%,2,1,2);
  708. 27*Y^3+54*X*Y^2+36*X^2*Y+Y+8*X^3+X+1;
  709. SUBSTPART(FACTOR(PIECE),%,[1,2,3,5]);
  710. 1/X+Y/X-1/Z;
  711. SUBSTPART(XTHRU(PIECE),%,[2,3]);
  712. SUBSTPART("+",%,1,0);
  713. RATSIMP((K^2*X^2-1)*(COS(X)+EPS)/(3*K+N[1])/(5*K-N[2]));
  714. FACTOR(%);
  715. SUBSTPART(RATSIMP(PIECE),%,1,[1,2]);
  716. -SUBSTPART(-PIECE,%,1,1);
  717. A+B/(X*(Y+(A+B)*X)+1);
  718. SUBSTPART(MULTTHRU(PIECE),%,1,2,1);&&
  719. SUM  SUM(I^2+2^I,I,0,N),SIMPSUM;
  720. SUM(3^(-I),I,1,INF),SIMPSUM;
  721. SUM(I^2,I,1,4)*SUM(1/I^2,I,1,INF),SIMPSUM;
  722. SUM(I^2,I,1,5);&&
  723. SYMMETRIC DECLARE(H,SYMMETRIC);
  724. H(X,Z,Y);&&
  725. SYNTAX  MATCHFIX("{","}");
  726. INFIX("|");
  727. {X|X>0};
  728. {X|X<2};
  729. INFIX(".U.")$
  730. INFIX(".I.")$
  731. %TH(4).U.%TH(3);
  732. %TH(5).U.%TH(4);
  733. {1,2,3}$
  734. {3,4,5}$
  735. %TH(2).U.%TH(2).U.%;
  736. INFIX(".U.",100,100)$
  737. INFIX(".I.",120,120)$
  738. %TH(5).U.%TH(5).U.%;
  739. REMOVE(".U.",OPERATOR)$
  740. ERRCATCH(%TH(7).U.%TH(3));
  741. REMOVE(["{","}",".I.",".U."],OPERATOR)$&&
  742. TAYLOR  TAYLOR(SQRT(1+A*X+SIN(X)),X,0,3);
  743. %^2;
  744. TAYLOR(SQRT(1+X),X,0,5);
  745. %^2;
  746. PRODUCT((X^I+1)^2.5,I,1,INF)/(X^2+1);
  747. TAYLOR(%,X,0,3),KEEPFLOAT;
  748. TAYLOR(1/LOG(1+X),X,0,3);
  749. TAYLOR(COS(X)-SEC(X),X,0,5);
  750. TAYLOR((COS(X)-SEC(X))^3,X,0,5);
  751. TAYLOR((COS(X)-SEC(X))^-3,X,0,5);
  752. TAYLOR(SQRT(1-K^2*SIN(X)^2),X,0,6);
  753. TAYLOR((1+X)^N,X,0,4);
  754. TAYLOR(SIN(X+Y),X,0,3,Y,0,3);
  755. TAYLOR(SIN(X+Y),[X,Y],0,3);
  756. TAYLOR(1/SIN(X+Y),X,0,3,Y,0,3);
  757. TAYLOR(1/SIN(X+Y),[X,Y],0,3);&&
  758. TAYTORAT  TAYLOR(1+X,[X,0,3]);
  759. 1/%;
  760. TAYLOR(1+X+Y+Z,[X,0,3],[Y,1,2],[Z,2,1]);
  761. 1/%;
  762. TAYLOR(1+X+Y+Z,[X,0,3],[Y,0,3],[Z,0,3]);
  763. 1/%;&&
  764. TELLRAT  10*(1+%I)/(3^(1/3)+%I);
  765. RATDISREP(RAT(%)),ALGEBRAIC;
  766. TELLRAT(A^2+A+1);
  767. A/(SQRT(2)+SQRT(3))+1/(A*SQRT(2)-1);
  768. RATDISREP(RAT(%)),ALGEBRAIC;
  769. TELLRAT(Y^2=X^2);&&
  770. TELLSIMP  MATCHDECLARE(X,FREEOF(%I))$
  771. %IARGS:FALSE$
  772. TELLSIMP(SIN(%I*X),%I*SINH(X));
  773. TRIGEXPAND(SIN(X+%I*Y));
  774. %IARGS:TRUE$
  775. ERRCATCH(0^0);
  776. TELLSIMP(0^0,1),SIMP:FALSE;
  777. 0^0;
  778. REMRULE("^",%th(2)[1]);
  779. TELLSIMP(SIN(X)^2,1-COS(X)^2)$
  780. (SIN(X)+1)^2;
  781. EXPAND(%);
  782. SIN(X)^2;
  783. KILL(RULES);
  784. MATCHDECLARE(A,TRUE)$
  785. TELLSIMP(SIN(A)^2,1-COS(A)^2)$
  786. SIN(Y)^2;
  787. KILL(RULES);&&
  788. TRIANGULARIZE  MATRIX([2,1-A,-5*B],[A,B,C]);
  789. TRIANGULARIZE(%);&&
  790. TRIG  SIN(%PI/12)+TAN(%PI/6);
  791. EV(%,NUMER);
  792. SIN(1);
  793. SIN(1),NUMER;
  794. BETA(1/2,2/5);
  795. EV(%,NUMER);
  796. DIFF(ATANH(SQRT(X)),X);
  797. FPPREC:25$
  798. SIN(0.5B0);
  799. COS(X)^2-SIN(X)^2;
  800. EV(%,X:%PI/3);
  801. DIFF(%TH(2),X);
  802. INTEGRATE(%TH(3),X);
  803. EXPAND(%);
  804. TRIGEXPAND(%);
  805. TRIGREDUCE(%);
  806. SECH(X)^2*SINH(X)*TANH(X)/COTH(X)^2 + COSH(X)^2*SECH(X)^2*TANH(X)/COTH(X)^2
  807.     + SECH(X)^2*TANH(X)/COTH(X)^2;
  808. TRIGSIMP(%);
  809. EV(SIN(X),EXPONENTIALIZE);
  810. TAYLOR(SIN(X)/X,X,0,4);
  811. EV(COS(X)^2-SIN(X)^2,SIN(X)^2=1-COS(X)^2);&&
  812. TRIGEXPAND  X+SIN(3*X)/SIN(X),TRIGEXPAND,EXPAND;
  813. TRIGEXPAND(SIN(10*X+Y));&&
  814. TRIGREDUCE  -SIN(X)^2+3*COS(X)^2+X;
  815. EXPAND(TRIGREDUCE(%));
  816. DECLARE(J,INTEGER,E,EVEN,O,ODD);
  817. SIN(X+(E+1/2)*%PI);
  818. SIN(X+(O+1/2)*%PI);&&
  819. UNORDER  A^2+B*X;
  820. ORDERGREAT(A);
  821. A^2+B*X;
  822. %-%TH(3);
  823. UNORDER();&&
  824. XTHRU  ((X+2)^20-2*Y)/(X+Y)^20+(X+Y)^-19-X/(X+Y)^20;
  825. XTHRU(%);&&
  826. ZEROEQUIV  ZEROEQUIV(SIN(2*X)-2*SIN(X)*COS(X),X);
  827. ZEROEQUIV(%E^X+X,X);
  828. ZEROEQUIV(LOG(A*B)-LOG(A)-LOG(B),A);
  829.