mirror of
https://github.com/PDP-10/its.git
synced 2026-02-17 13:17:18 +00:00
Eric Swenson
930
src/demo/manual.78
Normal file
930
src/demo/manual.78
Normal file
@@ -0,0 +1,930 @@
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/* This file is to be run by the EXAMPLE command, and may not
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otherwise work. */
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functions&& F(X):=X^2+Y;
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F(2);
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EV(F(2),Y:7);
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F(X):=SIN(X)^2+1;
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F(X+1);
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G(Y,Z):=F(Z)+3*Y;
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EV(G(2*Y+Z,-0.5),Y:7);
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H(N):=SUM(I*X^I,I,0,N);
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FUNCTIONS;
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T[N](X):=RATEXPAND(2*X*T[N-1](X)-T[N-2](X));
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T[0](X):=1$
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T[1](X):=X$
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T[4](Y);
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G[N](X):=SUM(EV(X),I,N,N+2);
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H(N,X):=SUM(EV(X),I,N,N+2);
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G[2](I^2);
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H(2,I^2);
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P[N](X):=RATSIMP(1/(2^N*N!)*DIFF((X^2-1)^N,X,N));
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Q(N,X):=RATSIMP(1/(2^N*N!)*DIFF((X^2-1)^N,X,N));
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P[2];
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P[2](Y+1);
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Q(2,Y+1);
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P[2](5);
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F[I,J](X,Y):=X^I+Y^J;
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G(FUN,A,B):=PRINT(FUN," applied to ",A," and ",B," is ",FUN(A,B))$
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G(F[2,1],SIN(%PI),2*C);
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arrays&& A[N]:=N*A[N-1];
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A[0]:1$
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A[5];
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A[N]:=N$
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A[6];
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A[4];
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lambda&& LAMBDA([X,Y,Z],X^2+Y^2+Z^2);
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%(1,2,A);
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"+"(1,2,A);
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lists&& [X^2,Y/3,-2];
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%[1]*X;
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[A,%TH(2),%];
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matrices&& M:MATRIX([A,0],[B,1]);
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M^2;
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M.M;
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M[1,1]*M;
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%-%TH(2)+1;
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M^^-1;
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[X,Y].M;
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MATRIX([A,B,C],[D,E,F],[G,H,I]);
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%^^2;
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equations&& X+1=Y^2;
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X-1=2*Y+1$
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%TH(2)+%;
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%TH(3)/Y;
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1/%;
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if&& FIB[N]:=IF N=1 OR N=2 THEN 1 ELSE FIB[N-1]+FIB[N-2];
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FIB[1]+FIB[2];
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FIB[3];
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FIB[5];
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ETA(MU,NU):=IF MU=NU THEN MU ELSE IF MU>NU THEN MU-NU ELSE MU+NU;
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ETA(5,6);
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ETA(ETA(7,7),ETA(1,2));
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IF NOT 5>=2 AND 6<=5 OR 4+1>3 THEN A ELSE B;
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block&& HESSIAN(F):=BLOCK([DFXX,DFXY,DFXZ,DFYY,DFYZ,DFZZ],
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DFXX:DIFF(F,X,2),DFXY:DIFF(F,X,1,Y,1),
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DFXZ:DIFF(F,X,1,Z,1),DFYY:DIFF(F,Y,2),
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DFYZ:DIFF(F,Y,1,Z,1),DFZZ:DIFF(F,Z,2),
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DETERMINANT(MATRIX([DFXX,DFXY,DFXZ],[DFXY,DFYY,DFYZ],
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[DFXZ,DFYZ,DFZZ])))$
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HESSIAN(X^3-3*A*X*Y*Z+Y^3);
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SUBST(1,Z,QUOTIENT(%,-54*A^2));
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F(X):=BLOCK([Y], LOCAL(A), Y:4, A[Y]:X, DISPLAY(A[Y]))$
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Y:2$
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A[Y+2]:0$
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F(9);
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A[Y+2];
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do&& FOR A:-3 THRU 26 STEP 7 DO LDISPLAY(A)$
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S:0$
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FOR I:1 WHILE I<=10 DO S:S+I;
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S;
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SERIES:1$
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TERM:EXP(SIN(X))$
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FOR P:1 UNLESS P>7 DO
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(TERM:DIFF(TERM,X)/P,
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SERIES:SERIES+SUBST(X=0,TERM)*X^P)$
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SERIES;
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POLY:0$
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FOR I:1 THRU 5 DO
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FOR J:I STEP -1 THRU 1 DO
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POLY:POLY+I*X^J$
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POLY;
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GUESS:-3.0$
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FOR I THRU 10 DO (GUESS:SUBST(GUESS,X,0.5*(X+10/X)),
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IF ABS(GUESS^2-10)<0.00005 THEN RETURN(GUESS));
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FOR COUNT:2 NEXT 3*COUNT THRU 20
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DO LDISPLAY(COUNT)$
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X:1000;
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THRU 10 WHILE X#0 DO X:0.5*(X+5/X)$
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X;
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REMVALUE(X);
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NEWTON(F,GUESS):=BLOCK([NUMER,Y],
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LOCAL(DF), NUMER:TRUE,
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DEFINE(DF(X),DIFF(F(X),X)),
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DO (Y:DF(GUESS), IF Y=0 THEN ERROR(
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"derivative at",GUESS,"is zero"),
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GUESS:GUESS-F(GUESS)/Y,
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IF ABS(F(GUESS))<5.0E-6 THEN RETURN(GUESS)))$
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SQR(X):=X^2-5.0$
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NEWTON(SQR,1000);
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FOR F IN [LOG, RHO, ATAN] DO LDISP(F(1.0))$
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EV(CONCAT(E,LINENUM-1),NUMER);
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evaluation&& DIFF(X*F(X),X);
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F(X):=SIN(X)$
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EV(%TH(2),DIFF);
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X;
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X:3$
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X;
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'X;
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F(X):=X^2;
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'F(2);
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EV(%,F);
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'(F(2));
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''%;
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SUM(I!,I,1,4);
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'SUM(I!,I,1,4);
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'INTEGRATE(F(X),X,A,B);
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FOR I THRU 5 DO S:S+I^2;
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S;
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EV(%,S:0);
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EV(%TH(2));
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'SUM(G(I),I,0,N);
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Z*%E^Z;
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EV(%,Z:X^2);
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SUBST(X^2,Z,%TH(3));
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A:%;
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A+1;
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KILL(A);
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A;
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DECLARE(INTEGRATE,NOUN)$
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INTEGRATE(Y^2,Y);
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''INTEGRATE(Y^2,Y);
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F(Y):=DIFF(Y*LOG(Y),Y,2);
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F(Y):=''(DIFF(Y*LOG(Y),Y,2));
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''(CONCAT(C,LINENUM-1));
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(X+Y)^3$
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DIFF(%,X);
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Y:X^2+1$
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''(CONCAT(C,LINENUM-2));
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exp&& EV(%E^X*SIN(X)^2,EXPONENTIALIZE);
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INTEGRATE(%,X);
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EV(%,DEMOIVRE);
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ANS:EV(%,RATEXPAND);
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EV(%,X:1,NUMER)-EV(%,X:0,NUMER);
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INTEGRATE(%E^X*SIN(X)^2,X);
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TRIGREDUCE(%);
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%-ANS;
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EV(SIN(X),%EMODE);
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trig&& SIN(%PI/12)+TAN(%PI/6);
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EV(%,NUMER);
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SIN(1);
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SIN(1),NUMER;
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BETA(1/2,2/5);
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EV(%,NUMER);
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DIFF(ATANH(SQRT(X)),X);
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FPPREC:25$
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SIN(0.5B0);
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COS(X)^2-SIN(X)^2;
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EV(%,X:%PI/3);
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DIFF(%TH(2),X);
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INTEGRATE(%TH(3),X);
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EXPAND(%);
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TRIGEXPAND(%);
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TRIGREDUCE(%);
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SECH(X)^2*SINH(X)*TANH(X)/COTH(X)^2 + COSH(X)^2*SECH(X)^2*TANH(X)/COTH(X)^2
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+ SECH(X)^2*TANH(X)/COTH(X)^2;
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TRIGSIMP(%);
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EV(SIN(X),EXPONENTIALIZE);
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TAYLOR(SIN(X)/X,X,0,4);
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EV(COS(X)^2-SIN(X)^2,SIN(X)^2=1-COS(X)^2);
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complex&& (SQRT(-4)+SQRT(2.25))^2;
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EXPAND(%);
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EXPAND(SQRT(2*%I));
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ev&& SIN(X)+COS(Y)+(W+1)^2+'DIFF(SIN(W),W);
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EV(%,SIN,EXPAND,DIFF,X=2,Y=1);
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EV(X+Y,X:A+Y,Y:2);
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'DIFF(Y^2+X*Y+X^2,X,2,Y,1);
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EV(%,DIFF);
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2*X-3*Y=3$
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-3*X+2*Y=-4$
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SOLVE([%TH(2),%]);
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EV(%TH(3),%);
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X+1/X>GAMMA(1/2);
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EV(%,NUMER,X=1/2);
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EV(%,PRED);
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zeroequiv&& ZEROEQUIV(SIN(2*X)-2*SIN(X)*COS(X),X);
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ZEROEQUIV(%E^X+X,X);
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ZEROEQUIV(LOG(A*B)-LOG(A)-LOG(B),A);
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expand&& (1/(X+Y)^4-3/(Y+Z)^3)^2;
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EXPAND(%,2,0);
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EXPAND(A.(B+C.(D+E)+F));
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EXPAND((X+1)^3);
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(X+1)^7;
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EXPAND(%);
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EXPAND(%TH(2),7,7);
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EV(A*(B+C)+A*(B+C)^2,EXPOP:1);
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ratexpand&& RATEXPAND((2*X-3*Y)^3);
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(X-1)/(X+1)^2+1/(X-1);
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EXPAND(%);
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RATEXPAND(%TH(2));
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ratsimp&& SIN(X/(X^2+X))=%E^((LOG(X)+1)^2-LOG(X)^2);
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RATSIMP(%);
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B*(A/B-X)+B*X+A;
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RATSIMP(%);
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((X-1)^(3/2)-(X+1)*SQRT(X-1))/SQRT((X-1)*(X+1));
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RATSIMP(%);
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EV(X^(A+1/A),RATSIMPEXPONS);
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radcan&& (LOG(X^2+X)-LOG(X))^A/LOG(X+1)^(A/2);
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RADCAN(%);
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LOG(A^(2*X)+2*A^X+1)/LOG(A^X+1);
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RADCAN(%);
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(%E^X-1)/(%E^(X/2)+1);
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RADCAN(%);
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multthru&& X/(X-Y)^2-1/(X-Y)-F(X)/(X-Y)^3;
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MULTTHRU((X-Y)^3,%);
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RATEXPAND(%);
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((A+B)^10*S^2+2*A*B*S+(A*B)^2)/(A*B*S^2);
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MULTTHRU(%);
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MULTTHRU(A.(B+C.(D+E)+F));
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xthru&& ((X+2)^20-2*Y)/(X+Y)^20+(X+Y)^-19-X/(X+Y)^20;
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XTHRU(%);
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partfrac&& 2/(X+2)-1/(X+1)-X/(X+1)^2;
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RATSIMP(%);
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PARTFRAC(%,X);
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factor&& FACTOR(2^63-1);
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FACTOR(Z^2*(X+2*Y)-4*X-8*Y);
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X^2*Y^2+2*X*Y^2+Y^2-X^2-2*X-1;
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DONTFACTOR:[X]$
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FACTOR(%/36/(Y^2+2*Y+1));
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FACTOR(%E^(3*X)+1);
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FACTOR(X^4+1,A^2-2);
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FACTOR(X^3+X^2*Y^2-X*Z^2-Y^2*Z^2);
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(X+2)/(X+3)/(X+B)/(X+C)^2;
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RATSIMP(%);
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PARTFRAC(%,X);
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MAP(FACOR,%);
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RATSIMP((X^5-1)/(X-1));
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SUBST(A,X,%);
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FACTOR(%TH(2),%);
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MAP(FACTOR,%TH(2)),POWERDISP;
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FACTOR(X^6+1);
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FACTOR(X^12+1);
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FACTOR(X^99+1);
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factorsum&& EV((X+1)*((U+V)^2+A*(W+Z)^2),EXPAND);
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FACTORSUM(%);
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sqfr&& SQFR(4*X^4+4*X^3-3*X^2-4*X-1);
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gfactor&& GFACTOR(X^4-1);
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partition&& PARTITION(2*A*X*F(X),X);
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PARTITION(A+B,X);
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logcontract&& 2*(A*LOG(X) + 2*A*LOG(Y));
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LOGCONTRACT(%);
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LOGCONTRACT(LOG(SQRT(X+1)+SQRT(X)) + LOG(SQRT(X+1)-SQRT(X)));
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rootscontract&& ROOTSCONMODE:FALSE$
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||||
ROOTSCONTRACT(X^(1/2)*Y^(3/2));
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ROOTSCONTRACT(X^(1/2)*Y^(1/4));
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ROOTSCONMODE:TRUE$
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ROOTSCONTRACT(X^(1/2)*Y^(1/4));
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ROOTSCONTRACT(X^(1/2)*Y^(1/3));
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ROOTSCONMODE:ALL$
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ROOTSCONTRACT(X^(1/2)*Y^(1/4));
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ROOTSCONTRACT(X^(1/2)*Y^(1/3));
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ROOTSCONMODE:FALSE$
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||||
ROOTSCONTRACT(SQRT(SQRT(X+1)+SQRT(X))*SQRT(SQRT(X+1)-SQRT(X)));
|
||||
ROOTSCONMODE:TRUE$
|
||||
ROOTSCONTRACT(SQRT(SQRT(5)+5)-5^(1/4)*SQRT(SQRT(5)+1));
|
||||
diff&& DIFF(SIN(X)+X^3+2*X^2,X);
|
||||
DIFF(SIN(X)*COS(X),X);
|
||||
DIFF(SIN(X)*COS(X),X,2);
|
||||
DERIVABBREV:TRUE$
|
||||
DIFF(EXP(F(X)),X,2);
|
||||
'INTEGRATE(F(X,Y),Y,G(X),H(X));
|
||||
DIFF(%,X);
|
||||
depends&& DEPENDS(A,X);
|
||||
DIFF(A.A,X);
|
||||
DEPENDS(F,[X,Y],[X,Y],T);
|
||||
DIFF(F,T);
|
||||
gradef&& DEPENDS(Y,X);
|
||||
GRADEF(F(X,Y),X^2,G(X,Y));
|
||||
DIFF(F(X,Y),X);
|
||||
GRADEF(J(N,Z),'DIFF(J(N,Z),N),
|
||||
J(N-1,Z)-N/Z*J(N,Z))$
|
||||
RATSIMP(DIFF(J(2,X),X,2));
|
||||
integrate&& INTEGRATE(SIN(X)^3,X);
|
||||
INTEGRATE(%E^X/(%E^X+2),X);
|
||||
INTEGRATE(1/(X*LOG(X)),X);
|
||||
INTEGRATE(SIN(2*X+3),X);
|
||||
INTEGRATE(%E^X*ERF(X),X);
|
||||
INTEGRATE(X/(X^3+1),X);
|
||||
DIFF(%,X);
|
||||
RATSIMP(%);
|
||||
INTEGRATE(X^(5/4)/(X+1)^(5/2),X,0,INF);
|
||||
GRADEF(Q(X),SIN(X^2));
|
||||
DIFF(LOG(Q(R(X))),X);
|
||||
INTEGRATE(%,X);
|
||||
risch&& RISCH(X^2*ERF(X),X);
|
||||
DIFF(%,X),RATSIMP;
|
||||
changevar&& 'INTEGRATE(%E^SQRT(A*Y),Y,0,4);
|
||||
CHANGEVAR(%,Y-Z^2/A,Z,Y);
|
||||
part&& X+Y/Z^2;
|
||||
PART(%,1,2,2);
|
||||
'INTEGRATE(F(X),X,A,B)+X;
|
||||
PART(%,1,1);
|
||||
X^2+2*X=Y^2;
|
||||
%+1;
|
||||
LHS(%);
|
||||
PART(%TH(2),2);
|
||||
PART(%,1);
|
||||
27*Y^3+54*X*Y^2+36*X^2*Y+Y+8*X^3+X+1;
|
||||
PART(%,2,[1,3]);
|
||||
SQRT(PIECE/54);
|
||||
inpart&& X+Y+W*Z;
|
||||
INPART(%,3,2);
|
||||
PART(%TH(2),1,2);
|
||||
'LIMIT(F(X)^G(X+1),X,0,MINUS);
|
||||
INPART(%,1,2);
|
||||
nounify&& 'LIMIT(F(X)^G(X+1),X,0,MINUS);
|
||||
IS(INPART(%,0)=NOUNIFY(LIMIT));
|
||||
dpart&& DPART(X+Y/Z^2,1,2,1);
|
||||
EXPAND((B+A)^4);
|
||||
(B+A)^2*(Y+X)^2;
|
||||
EXPAND(%);
|
||||
%TH(3)/%;
|
||||
FACTOR(%);
|
||||
DPART(%TH(2),2,4);
|
||||
PART(%TH(3),2,4);
|
||||
subst&& SUBST(A,X+Y,X+(X+Y)^2+Y);
|
||||
SUBST(-%I,%I,A+B*%I);
|
||||
EV(A+B*%I,%I:-%I);
|
||||
SUBST(X,Y,X+Y);
|
||||
SUBST(X=0,DIFF(SIN(X),X));
|
||||
DIFF(SIN(X),X),X=0;
|
||||
INTEGRATE(X^I,X),I=-1;
|
||||
SUBST(-1,I,INTEGRATE(X^I,X));
|
||||
MATRIX([A,B],[C,D]);
|
||||
SUBST("[",MATRIX,%);
|
||||
ratsubst&& RATSUBST(A,X*Y^2,X^4*Y^8+X^4*Y^3);
|
||||
1 + COS(X) + COS(X)^2 + COS(X)^3 + COS(X)^4;
|
||||
RATSUBST(1-SIN(X)^2,COS(X)^2,%);
|
||||
RATSUBST(1-COS(X)^2,SIN(X)^2,SIN(X)^4);
|
||||
substpart&& 1/(X^2+2);
|
||||
SUBSTPART(3/2,%,2,1,2);
|
||||
27*Y^3+54*X*Y^2+36*X^2*Y+Y+8*X^3+X+1;
|
||||
SUBSTPART(FACTOR(PIECE),%,[1,2,3,5]);
|
||||
1/X+Y/X-1/Z;
|
||||
SUBSTPART(XTHRU(PIECE),%,[2,3]);
|
||||
SUBSTPART("+",%,1,0);
|
||||
RATSIMP((K^2*X^2-1)*(COS(X)+EPS)/(3*K+N[1])/(5*K-N[2]));
|
||||
FACTOR(%);
|
||||
SUBSTPART(RATSIMP(PIECE),%,1,[1,2]);
|
||||
-SUBSTPART(-PIECE,%,1,1);
|
||||
A+B/(X*(Y+(A+B)*X)+1);
|
||||
SUBSTPART(MULTTHRU(PIECE),%,1,2,1);
|
||||
substinpart&& X.'DIFF(F(X),X,2);
|
||||
SUBSTINPART(D^2,%,2);
|
||||
SUBSTINPART(F1,F[1](X+1),0);
|
||||
atvalue&& ATVALUE(F(X,Y),[X=0,Y=1],A^2)$
|
||||
ATVALUE('DIFF(F(X,Y),X),X=0,Y+1);
|
||||
PRINTPROPS(ALL,ATVALUE);
|
||||
DIFF(4*F(X,Y)^2-U(X,Y)^2,X);
|
||||
AT(%,[X=0,Y=1]);
|
||||
at&& ATVALUE(F(X,Y),[X=0,Y=1],A^2);
|
||||
ATVALUE('DIFF(F(X,Y),X),X=0,Y+1);
|
||||
PRINTPROPS(ALL,ATVALUE);
|
||||
DIFF(4*F(X,Y)^2-U(X,Y)^2,X);
|
||||
AT(%,[X=0,Y=1]);
|
||||
listofvars&& LISTOFVARS(F(X[1]+Y)/G^(2+A));
|
||||
coeff&& COEFF(2*A*TAN(X)+TAN(X)+B=5*TAN(X)+3,TAN(X));
|
||||
COEFF(Y+X*%E^X+1,X,0);
|
||||
ratcoeff&& A*X+B*X+5$
|
||||
RATCOEF(%,A+B);
|
||||
bothcoeff&& ISLINEAR(EXP,VAR):=BLOCK([C],
|
||||
C:BOTHCOEF(RAT(EXP,VAR),VAR),
|
||||
IS(FREEOF(VAR,C) AND C[1]#0))$
|
||||
ISLINEAR((R^2-(X-R)^2)/X,X);
|
||||
isolate&& (A+B)^4*(1+X*(2*X+(C+D)^2));
|
||||
ISOLATE(%,X);
|
||||
RATEXPAND(%)$
|
||||
EV(%);
|
||||
(A+B)*(X+A+B)^2*%E^(X^2+A*X+B);
|
||||
ISOLATE(%,X),EXPTISOLATE:TRUE;
|
||||
pickapart&& INTEGRATE(1/(X^3+2),X)$
|
||||
PICKAPART(%,1);
|
||||
numfactor&& GAMMA(7/2);
|
||||
NUMFACTOR(%);
|
||||
derivdegree&& 'DIFF(Y,X,2)+'DIFF(Y,Z,3)*2+'DIFF(Y,X)*X^2;
|
||||
DERIVDEGREE(%,Y,X);
|
||||
realpart&& (%I*V+U)/(F+%I*E)+%E^(%I*ALPHA);
|
||||
REALPART(%);
|
||||
polarform&& RECTFORM(SIN(2*%I+X));
|
||||
POLARFORM(%);
|
||||
RECTFORM(LOG(3+4*%I));
|
||||
POLARFORM(%);
|
||||
RECTFORM((2+3.5*%I)^0.25),NUMER;
|
||||
POLARFORM(%);
|
||||
delete&& DELETE(SIN(X),X+SIN(X)+Y);
|
||||
nroots&& X^10-2*X^4+1/2;
|
||||
NROOTS(%,-6,9.1);
|
||||
realroots&& REALROOTS(X^5-X-1,5.0E-6);
|
||||
%[1],FLOAT;
|
||||
X^5-X-1,%;
|
||||
allroots&& (2*X+1)^3=13.5*(X^5+1);
|
||||
ALLROOTS(%);
|
||||
linsolve&& X+Z=Y$
|
||||
2*A*X-Y=2*A^2$
|
||||
Y-2*Z=2$
|
||||
LINSOLVE([%TH(3),%TH(2),%],[X,Y,Z]),GLOBALSOLVE;
|
||||
algsys&& F1:2*X*(1-L1)-2*(X-1)*L2$
|
||||
F2:L2-L1$
|
||||
F3:L1*(1-X**2-Y)$
|
||||
F4:L2*(Y-(X-1)**2)$
|
||||
ALGSYS([F1,F2,F3,F4],[X,Y,L1,L2]);
|
||||
F1:X**2-Y**2$
|
||||
F2:X**2-X+2*Y**2-Y-1$
|
||||
ALGSYS([F1,F2],[X,Y]);
|
||||
solve&& SOLVE(ASIN(COS(3*X))*(F(X)-1),X);
|
||||
SOLVE(5^F(X)=125,F(X)),SOLVERADCAN;
|
||||
[4*X^2-Y^2=12,X*Y-X=2];
|
||||
SOLVE(%,[X,Y]);
|
||||
SOLVE(X^3+A*X+1,X);
|
||||
SOLVE(X^3-1);
|
||||
SOLVE(X^6-1);
|
||||
EV(X^6-1,%[1]);
|
||||
EXPAND(%);
|
||||
X^2-1;
|
||||
SOLVE(%,X);
|
||||
%TH(2),%[1];
|
||||
entermatrix&& ENTERMATRIX(2,1);
|
||||
genmatrix&& H[I,J]:=1/(I+J-1)$
|
||||
GENMATRIX(H,3,3);
|
||||
augcoefmatrix&& [2*X-(A-1)*Y=5*B,A*X+B*Y+C=0]$
|
||||
AUGCOEFMATRIX(%,[X,Y]);
|
||||
echelon&& MATRIX([2,1-A,-5*B],[A,B,C]);
|
||||
ECHELON(%);
|
||||
triangularize&& MATRIX([2,1-A,-5*B],[A,B,C]);
|
||||
TRIANGULARIZE(%);
|
||||
rank&& MATRIX([2,1-A,-5*B],[A,B,C]);
|
||||
RANK(%);
|
||||
charpoly&& A:MATRIX([3,1],[2,4]);
|
||||
EXPAND(CHARPOLY(A,LAMBDA));
|
||||
(PROGRAMMODE:TRUE,SOLVE(%));
|
||||
MATRIX([X1],[X2]);
|
||||
EV(A.%-LAMBDA*%,%TH(2)[1]);
|
||||
%[1,1]=0;
|
||||
X1^2+X2^2=1;
|
||||
SOLVE([%TH(2),%],[X1,X2]);
|
||||
MATRIX([X^3,X^2,X,1],[Y^3,Y^2,X,1],[Z^3,Z^2,Z,1],[W^3,W^2,W,1]);
|
||||
DETERMINANT(%);
|
||||
FACTOR(%);
|
||||
dotscrules&& DECLARE(L,SCALAR,[M1,M2,M3],NONSCALAR);
|
||||
EXPAND((1-L*M1).(1-L*M2).(1-L*M3));
|
||||
%,DOTSCRULES;
|
||||
RAT(%,L);
|
||||
rat&& RAT(X^2);
|
||||
DIFF(F(%),X);
|
||||
((X-2*Y)^4/(X^2-4*Y^2)^2+1)*(Y+A)*(2*Y+X)/(4*Y^2+X^2);
|
||||
RAT(%,Y,A,X);
|
||||
(X+3)^20;
|
||||
RAT(%);
|
||||
DIFF(%,X);
|
||||
FACTOR(%);
|
||||
ratweight&& RATWEIGHT(A,1,B,1);
|
||||
RAT(A+B+1);
|
||||
%^2;
|
||||
EV(%TH(2)^2,RATWTLVL:1);
|
||||
horner&& 1.0E-20*X^2-5.5*X+5.2E20;
|
||||
HORNER(%,X),KEEPFLOAT);
|
||||
EV(%,X=1.0E20);
|
||||
divide&& DIVIDE(X+Y,X-Y,X);
|
||||
DIVIDE(X+Y,X-Y);
|
||||
content&& CONTENT(2*X*Y+4*X^2*Y^2,Y);
|
||||
resultant&& RESULTANT(A*Y+X^2+1,Y^2+X*Y+B,X);
|
||||
ratdiff&& (4*X^3+10*X-11)/(X^5+5);
|
||||
MOD(%),MODULUS:3;
|
||||
RATDIFF(%TH(3),X);
|
||||
tellrat&& 10*(1+%I)/(3^(1/3)+%I);
|
||||
RATDISREP(RAT(%)),ALGEBRAIC;
|
||||
TELLRAT(A^2+A+1);
|
||||
A/(SQRT(2)+SQRT(3))+1/(A*SQRT(2)-1);
|
||||
RATDISREP(RAT(%)),ALGEBRAIC;
|
||||
TELLRAT(Y^2=X^2);
|
||||
taytorat&& TAYLOR(1+X,[X,0,3]);
|
||||
1/%;
|
||||
TAYLOR(1+X+Y+Z,[X,0,3],[Y,1,2],[Z,2,1]);
|
||||
1/%;
|
||||
TAYLOR(1+X+Y+Z,[X,0,3],[Y,0,3],[Z,0,3]);
|
||||
1/%;
|
||||
sum&& SUM(I^2+2^I,I,0,N),SIMPSUM;
|
||||
SUM(3^(-I),I,1,INF),SIMPSUM;
|
||||
SUM(I^2,I,1,4)*SUM(1/I^2,I,1,INF),SIMPSUM;
|
||||
SUM(I^2,I,1,5);
|
||||
product&& PRODUCT(X+I*(I+1)/2,I,1,4);
|
||||
limit&& LIMIT(X*LOG(X),X,0,PLUS);
|
||||
LIMIT((1+X)^(1/X),X,0);
|
||||
LIMIT(%E^X/X,X,INF);
|
||||
LIMIT(SIN(1/X),X,0);
|
||||
residue&& RESIDUE(S/(S^2+A^2),S,A*%I);
|
||||
RESIDUE(SIN(A*X)/X^4,X,0);
|
||||
taylor&& TAYLOR(SQRT(1+A*X+SIN(X)),X,0,3);
|
||||
%^2;
|
||||
TAYLOR(SQRT(1+X),X,0,5);
|
||||
%^2;
|
||||
PRODUCT((X^I+1)^2.5,I,1,INF)/(X^2+1);
|
||||
TAYLOR(%,X,0,3),KEEPFLOAT;
|
||||
TAYLOR(1/LOG(1+X),X,0,3);
|
||||
TAYLOR(COS(X)-SEC(X),X,0,5);
|
||||
TAYLOR((COS(X)-SEC(X))^3,X,0,5);
|
||||
TAYLOR((COS(X)-SEC(X))^-3,X,0,5);
|
||||
TAYLOR(SQRT(1-K^2*SIN(X)^2),X,0,6);
|
||||
TAYLOR((1+X)^N,X,0,4);
|
||||
TAYLOR(SIN(X+Y),X,0,3,Y,0,3);
|
||||
TAYLOR(SIN(X+Y),[X,Y],0,3);
|
||||
TAYLOR(1/SIN(X+Y),X,0,3,Y,0,3);
|
||||
TAYLOR(1/SIN(X+Y),[X,Y],0,3);
|
||||
deftaylor&& DEFTAYLOR(F(X),X^2+SUM(X^I/(2^I*I!^2),I,4,INF));
|
||||
TAYLOR(%E^SQRT(F(X)),X,0,4);
|
||||
powerseries&& POWERSERIES(LOG(SIN(X)/X),X,0);
|
||||
trigexpand&& X+SIN(3*X)/SIN(X),TRIGEXPAND,EXPAND;
|
||||
TRIGEXPAND(SIN(10*X+Y));
|
||||
trigreduce&& -SIN(X)^2+3*COS(X)^2+X;
|
||||
EXPAND(TRIGREDUCE(%));
|
||||
DECLARE(J,INTEGER,E,EVEN,O,ODD);
|
||||
SIN(X+(E+1/2)*%PI);
|
||||
SIN(X+(O+1/2)*%PI);
|
||||
optimize&& DIFF(EXP(X^2+Y)/(X+Y),X,2);
|
||||
OPTIMIZE(%);
|
||||
laplace&& LAPLACE(%E^(2*T+A)*SIN(T)*T,T,S);
|
||||
ilt&& 'INTEGRATE(SINH(A*X)*F(T-X),X,0,T)+B*F(T)=T^2;
|
||||
LAPLACE(%,T,S);
|
||||
LINSOLVE([%],['LAPLACE(F(T),T,S)]);
|
||||
ILT(EV(%[1]),S,T);
|
||||
minfactorial&& N!/(N+1)!;
|
||||
MINFACTORIAL(%);
|
||||
factcomb&& (N+1)^B*N!^B;
|
||||
FACTCOMB(%);
|
||||
qunit&& QUNIT(17);
|
||||
EXPAND(%*(SQRT(17)-4));
|
||||
cf&& CF([1,2,-3]+[1,-2,1]);
|
||||
CFDISREP(%);
|
||||
CFLENGTH:4$
|
||||
CF(SQRT(3));
|
||||
CFEXPAND(%);
|
||||
EV(%[1,2]/%[2,2],NUMER);
|
||||
cfdisrep&& CF([1,2,-3]+[1,-2,1]);
|
||||
CFDISREP(%);
|
||||
cfexpand&& CFLENGTH:4$
|
||||
CF(SQRT(3));
|
||||
CFEXPAND(%);
|
||||
EV(%[1,2]/%[2,2],NUMER);
|
||||
featurep&& DECLARE(J,EVEN)$
|
||||
FEATUREP(J,INTEGER);
|
||||
map&& MAP(F,X+A*Y+B*Z);
|
||||
MAP(LAMBDA([U],PARTFRAC(U,X)),X/(X^3+4*X^2+5*X+2));
|
||||
MAP(RATSIMP, X/(X^2+X)+(Y^2+Y)/Y);
|
||||
MAP("=",[A,B],[-0.5,3,2.5]);
|
||||
fullmap&& FULLMAP(G,A+B*C);
|
||||
MAP(G,A+B*C);
|
||||
fullmapl&& FULLMAPL("+",[3,[4,5]],[[A,1],[0,-1.5]]);
|
||||
scanmap&& (A^2+2*A+1)*Y+X^2;
|
||||
SCANMAP(FACTOR,%);
|
||||
SCANMAP(FACTOR,%TH(2)AND(EXP));
|
||||
U*V^(A*X+B)+C;
|
||||
SCANMAP('F,%);
|
||||
append&& APPEND([Y+X,0,-3.2],[2.5E20,X]);
|
||||
reverse&& UNION(X,Y):=IF X=[] THEN Y ELSE
|
||||
IF MEMBER(T:FIRST(X),Y) THEN UNION(REST(X),Y)
|
||||
ELSE CONS(T,UNION(REST(X),Y)$
|
||||
UNION([A,B,1,1/2,X^2],[-X^2,A,Y,1/2]);
|
||||
BERNPOLY(X,5);
|
||||
MAPLIST(NUMFACTOR,%);
|
||||
APPLY(MIN,%);
|
||||
display&& DISPLAY(B[1,2]);
|
||||
reveal&& INTEGRATE(1/(X^3+2),X)$
|
||||
REVEAL(%,2);
|
||||
REVEAL(%TH(2),3);
|
||||
catch&& G(L):=CATCH(MAP(LAMBDA([X],IF X<0 THEN THROW(X) ELSE F(X)),L))$
|
||||
G([1,2,3,7]);
|
||||
G([1,2,-3,7]);
|
||||
unorder&& A^2+B*X;
|
||||
ORDERGREAT(A);
|
||||
A^2+B*X;
|
||||
%-%TH(3);
|
||||
UNORDER();
|
||||
arrayinfo&& B[1,X]:1$
|
||||
ARRAY(F,2,3);
|
||||
ARRAYINFO(B);
|
||||
ARRAYINFO(F);
|
||||
properties&& PROPERTIES(CONS);
|
||||
ASSUME(VAR1>0);
|
||||
PROPERTIES(VAR1);
|
||||
VAR2:2$
|
||||
PROPERTIES(VAR2);
|
||||
printprops&& GRADEF(R,X,X/R)$
|
||||
GRADEF(R,Y,Y/R)$
|
||||
PRINTPROPS(R,ATOMGRAD);
|
||||
PROPVARS(ATOMGRAD);
|
||||
propvars&& GRADEF(R,X,X/R)$
|
||||
GRADEF(R,Y,Y/R)$
|
||||
PRINTPROPS(R,ATOMGRAD);
|
||||
PROPVARS(ATOMGRAD);
|
||||
get&& PUT(%E,TRANSCENDENTAL,TYPE);
|
||||
PUT(%PI,TRANSCENDENTAL,TYPE)$
|
||||
PUT(%I,ALGEBRAIC,TYPE)$
|
||||
TYPEOF(X):=BLOCK([Q], IF NUMBERP(X)
|
||||
THEN RETURN(ALGEBRAIC),
|
||||
IF NOT ATOM(X)
|
||||
THEN RETURN(MAPLIST(TYPEOF,X)),
|
||||
Q:GET(X,TYPE), IF Q=FALSE THEN
|
||||
ERROR("NOT NUMERIC") ELSE Q)$
|
||||
ERRCATCH(TYPEOF(2*%E+X*%PI));
|
||||
TYPEOF(2*%E+%PI);
|
||||
is&& IS(X^2>=2*X-1);
|
||||
IS(EQUAL(Y^3,1) OR LOG(X)>0);
|
||||
ASSUME(A>1);
|
||||
IS(LOG(LOG(A+1)+1)>0 AND A^2+1>2*A);
|
||||
freeof&& FREEOF(Y,SIN(X+2*Y));
|
||||
FREEOF(COS(Y),"*",SIN(Y)+COS(X));
|
||||
matchdeclare&& MATCHDECLARE(A,TRUE)$
|
||||
TELLSIMP(SIN(A)^2,1-COS(A)^2)$
|
||||
SIN(Y)^2;
|
||||
KILL(RULES);
|
||||
NONZEROANDFREEOF(X,E):=IS(E#0 AND FREEOF(X,E));
|
||||
MATCHDECLARE(A,NONZEROANDFREEOF(X),B,FREEOF(X));
|
||||
DEFMATCH(LINEAR,A*X+B,X);
|
||||
LINEAR(3*Z+(Y+1)*Z+Y**2,Z);
|
||||
MATCHDECLARE([A,F],TRUE);
|
||||
CONSTINTERVAL(L,H):=CONSTANTP(H-L)$
|
||||
MATCHDECLARE(B,CONSTINTERVAL(A))$
|
||||
MATCHDECLARE(X,ATOM)$
|
||||
BLOCK(REMOVE(INTEGRATE,OUTATIVE),
|
||||
DEFMATCH(CHECKLIMITS,'INTEGRATE(F,X,A,B)),
|
||||
DECLARE(INTEGRATE,OUTATIVE))$
|
||||
'INTEGRATE(SIN(T),T,X+%PI,X+2*%PI)$
|
||||
CHECKLIMITS(%);
|
||||
'INTEGRATE(SIN(T),T,0,X)$
|
||||
CHECKLIMITS(%);
|
||||
tellsimp&& MATCHDECLARE(X,FREEOF(%I))$
|
||||
%IARGS:FALSE$
|
||||
TELLSIMP(SIN(%I*X),%I*SINH(X));
|
||||
TRIGEXPAND(SIN(X+%I*Y));
|
||||
%IARGS:TRUE$
|
||||
ERRCATCH(0^0);
|
||||
TELLSIMP(0^0,1),SIMP:FALSE;
|
||||
0^0;
|
||||
REMRULE("^","^RULE1");
|
||||
TELLSIMP(SIN(X)^2,1-COS(X)^2)$
|
||||
(SIN(X)+1)^2;
|
||||
EXPAND(%);
|
||||
SIN(X)^2;
|
||||
KILL(RULES);
|
||||
MATCHDECLARE(A,TRUE)$
|
||||
TELLSIMP(SIN(A)^2,1-COS(A)^2)$
|
||||
SIN(Y)^2;
|
||||
KILL(RULES);
|
||||
defmatch&& NONZEROANDFREEOF(X,E):=IS(E#0 AND FREEOF(X,E));
|
||||
MATCHDECLARE(A,NONZEROANDFREEOF(X),B,FREEOF(X));
|
||||
DEFMATCH(LINEAR,A*X+B,X);
|
||||
LINEAR(3*Z+(Y+1)*Z+Y**2,Z);
|
||||
MATCHDECLARE([A,F],TRUE);
|
||||
CONSTINTERVAL(L,H):=CONSTANTP(H-L)$
|
||||
MATCHDECLARE(B,CONSTINTERVAL(A))$
|
||||
MATCHDECLARE(X,ATOM)$
|
||||
BLOCK(REMOVE(INTEGRATE,OUTATIVE),
|
||||
DEFMATCH(CHECKLIMITS,'INTEGRATE(F,X,A,B)),
|
||||
DECLARE(INTEGRATE,OUTATIVE))$
|
||||
'INTEGRATE(SIN(T),T,X+%PI,X+2*%PI)$
|
||||
CHECKLIMITS(%);
|
||||
'INTEGRATE(SIN(T),T,0,X)$
|
||||
CHECKLIMITS(%);
|
||||
let&& MATCHDECLARE([A,A1,A2],TRUE);
|
||||
ONELESS(X,Y):=IS(X=Y-1)$
|
||||
LET(A1*A2!,A1!,ONELESS,A2,A1);
|
||||
LET(A1!/A1,(A1-1)!),LETRAT;
|
||||
LETSIMP(N*M!*(N-1)!/M),LETRAT;
|
||||
LET(SIN(A)^2,1-COS(A)^2);
|
||||
SIN(X)^4;
|
||||
LETSIMP(%);
|
||||
letrules&& MATCHDECLARE([A,A1,A2],TRUE);
|
||||
ONELESS(X,Y):=IS(X=Y-1)$
|
||||
LET(A1*A2!,A1!,ONELESS,A2,A1);
|
||||
LET(A1!/A1,(A1-1)!),LETRAT;
|
||||
LETSIMP(N*M!*(N-1)!/M),LETRAT;
|
||||
LET(SIN(A)^2,1-COS(A)^2);
|
||||
SIN(X)^4;
|
||||
LETSIMP(%);
|
||||
poissimp&& PFEFORMAT:TRUE$
|
||||
POISSIMP(SIN(X)^2);
|
||||
(2*A^2-B)*COS(X+2*Y)-(A*B+5)*SIN(U-4*X);
|
||||
POISEXPT(%,2)$
|
||||
PRINTPOIS(%);
|
||||
POISINT(%TH(2),Y)$
|
||||
POISSIMP(%);
|
||||
POISSIMP(SIN(X)^5+COS(X)^5);
|
||||
PFEFORMAT:FALSE$
|
||||
tensor&& LOADFILE(ETENSR,FASL,DSK,SHARE);
|
||||
setup&& SETUP();
|
||||
christof&& CHRISTOF(MCS);
|
||||
riccicom&& RICCICOM(TRUE);
|
||||
riemann&& RIEMANN(TRUE);
|
||||
rinvariant&& RINVARIANT();
|
||||
dscalar&& DEPENDS(FIELD,R);
|
||||
DSCALAR(FIELD);
|
||||
riemann&& G(L1,L2):=BLOCK([A,B],IF L2=[]
|
||||
THEN [A:L1[1],B:L1[2],RETURN(E(L1,[])*(1+2*L*P([],[]))-4*L*P(L1,[]))],
|
||||
A:L2[1],B:L2[2],E([],L2)*(1-2*L*P([],[]))+4*L*P([],L2))$
|
||||
METRIC:G$
|
||||
DEFCON(G);
|
||||
DECLARE(E,CONSTANT);
|
||||
DEFCON(E);
|
||||
DEFCON(E,E,DELTA)$
|
||||
RIEMANN([I,J,K],[J])$
|
||||
SHOW(%);
|
||||
UNDIFF(%TH(2))$
|
||||
EV(%,CHR2,DIFF)$
|
||||
RATWEIGHT(L,1)$
|
||||
RATEXPAND(%TH(2)),RATWTLVL:1,TAKEGCD;
|
||||
CONTRACT(%)$
|
||||
SHOW(%);
|
||||
can&& P([I,J,S,V],[M,N,Q],V)*P1([Q,T],[R,S])
|
||||
*P2([R,L,M,N,U],[I,J,K]) +
|
||||
P2([L,N,U],[R,M,I,J,K])
|
||||
*P([M,I,J,V],[N,Q,S],V)*P1([R,S,Q,T],[])$
|
||||
SHOW(%);
|
||||
SHOW(RENAME(%TH(2)));
|
||||
SHOW(CAN(%TH(3)));
|
||||
BATCH(ITENSR,DEMO,DSK,SHARE);
|
||||
/* DEMONSTRATION OF MACSYMAS INDICIAL MANIPULATION
|
||||
OF SYMMETRIC TENSORS. WE WILL SHOW THAT THE COVARIANT
|
||||
DERIVATIVE OF THE COVARIANT FORM OF THE METRIC TENSOR
|
||||
IS ZERO */
|
||||
|
||||
/* LOAD IN INDICIAL TENSOR MANIPULATION PACKAGE */
|
||||
|
||||
LOADFILE(ITENSR,FASL,DSK,SHARE)$
|
||||
/* STATE THAT G IS THE METRIC AND THAT IT CONTRACTS
|
||||
WITH ITSELF TO FORM THE KRONECKER DELTA */
|
||||
|
||||
DEFCON(G)$
|
||||
DEFCON(G,G,DELTA)$
|
||||
METRIC:G$
|
||||
/* SHOW THE TIMES OF THE COMPUTATIONS */
|
||||
|
||||
TIME:TRUE$
|
||||
/* OBTAIN COVARIANT DERIVATIVE OF METRIC
|
||||
AND DISPLAY IT */
|
||||
|
||||
E:COVDIFF(G([I,J]),K)$
|
||||
SHOW(%)$
|
||||
/* REPLACE CHRISTOFFEL SYMBOLS OF SECOND
|
||||
KIND BY THEIR VALUE */
|
||||
|
||||
E:EV(E,CHR2)$
|
||||
SHOW(%)$
|
||||
/* EXPAND OUT COMPLETELY */
|
||||
|
||||
E:EXPAND(E)$
|
||||
SHOW(%)$
|
||||
/* CONTRACT INDICES MAKING USE OF ALL RULES
|
||||
AND THE RESULT IS 0 */
|
||||
|
||||
CONTRACT(E);
|
||||
ode2&& X^2*'DIFF(Y,X) + 3*X*Y = SIN(X)/X;
|
||||
SOLN1:ODE2(%,Y,X);
|
||||
IC1(SOLN1,X=%PI,Y=0);
|
||||
'DIFF(Y,X,2) + Y*'DIFF(Y,X)^3 = 0;
|
||||
SOLN2:ODE2(%,Y,X);
|
||||
RATSIMP(IC2(SOLN2,X=0,Y=0,'DIFF(Y,X)=2));
|
||||
BC2(SOLN2,X=0,Y=1,X=1,Y=3);
|
||||
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;
|
||||
EQ1:K^2+L^2=1;
|
||||
EQ2:N^2-M^2=1;
|
||||
SCSIMP(EXP,EQ1,EQ2);
|
||||
EXQ:(K1*K4-K1*K2-K2*K3)/K3^2;
|
||||
EQ3:K1*K4-K2*K3=0;
|
||||
EQ4:K1*K2+K3*K4=0;
|
||||
SCSIMP(EXQ,EQ3,EQ4);
|
||||
eliminate&& EXP1:2*X^2+Y*X+Z;
|
||||
EXP2:3*X+5*Y-Z-1;
|
||||
EXP3:Z^2+X-Y^2+5;
|
||||
ELIMINATE([EXP3,EXP2,EXP1],[Y,Z]);
|
||||
desolve&& EQN1:'DIFF(F(X),X)='DIFF(G(X),X)+SIN(X);
|
||||
EQN2:'DIFF(G(X),X,2)='DIFF(F(X),X)-COS(X);
|
||||
ATVALUE('DIFF(G(X),X),X=0,A);
|
||||
ATVALUE(F(X),X=0,1);
|
||||
DESOLVE([EQN1,EQN2],[F(X),G(X)]);
|
||||
/* VERIFICATION */
|
||||
[EQN1,EQN2],%,DIFF;
|
||||
break&& ROOT(F,V):=BLOCK([VAL,FUN,DER],DER:DIFF(F,V),VAL:0,
|
||||
WHILE(ABS(FUN:SUBST(VAL,V,F))<5.0E-7
|
||||
DO VAL:VAL-FUN/DER:SUBST(VAL,V,DER),
|
||||
VAL)$
|
||||
SIN(%PI*X)-%PI*(X-1),NUMER$
|
||||
ROOT(%,X);
|
||||
DEBUGMODE(TRUE);
|
||||
ROOT(%TH(3),X),DEBUG;
|
||||
TRACE(SUBST);
|
||||
ROOT(%TH(5),X);
|
||||
ROOT(F,V):=BLOCK([VAL,FUN,DER],DER:DIFF(F,V,1),VAL:0,
|
||||
TEST,FUN:SUBST(VAL,V,F),IF ABS(FUN)<5.0E-7 THEN
|
||||
RETURN(VAL),DER:SUBST(VAL,V,DER),IF ABS(DER)<5.E-8
|
||||
THEN ERROR("Derivative is zero"),VAL:VAL-FUN/DER,
|
||||
GO(TEST))$
|
||||
UNTRACE();
|
||||
ERRCATCH(ROOT(%TH(8),X));
|
||||
syntax&& MATCHFIX("{","}");
|
||||
INFIX("|");
|
||||
{X|X>0};
|
||||
{X|X<2};
|
||||
INFIX(".U.")$
|
||||
INFIX(".I.")$
|
||||
%TH(4).U.%TH(3);
|
||||
%TH(5).U.%TH(4);
|
||||
{1,2,3}$
|
||||
{3,4,5}$
|
||||
%TH(2).U.%TH(2).U.%;
|
||||
INFIX(".U.",100,100)$
|
||||
INFIX(".I.",120,120)$
|
||||
%TH(5).U.%TH(5).U.%;
|
||||
REMOVE(".U.",OPERATOR)$
|
||||
ERRCATCH(%TH(7).U.%TH(3));
|
||||
solder&& BATCH(SOLDER,DEMO,DSK,DEMO);
|
||||
/* THE FOLLOWING ROUTINE RETURNS THE HOMOG.-PART SOLN.
|
||||
TO 2ND ORDER LINEAR DIFF'L EQNS. WITH CONST. COEFFS. */
|
||||
|
||||
MATCHDECLARE([B,C],RATNUMP)$
|
||||
MATCHDECLARE(F,FREEOF(U))$
|
||||
ALIAS(D,DIFF)$
|
||||
DEFMATCH(SOLDE,'D(U,X,2) + B*'D(U,X) + C*U = F,U,X)$
|
||||
SOLDER(EQN,U,X) :=
|
||||
BLOCK([B,C,F,DISC,R1,R2,ALPHA,BETA],
|
||||
IF SOLDE(EQN,U,X) = FALSE THEN RETURN(FALSE),
|
||||
DISC: B^2 - 4*C, ALPHA: -B/2,
|
||||
IF DISC=0 THEN RETURN(%E^(ALPHA*X)*(A1+A2*X)),
|
||||
BETA: SQRT(DISC)/2,
|
||||
IF DISC > 0
|
||||
THEN (R1: ALPHA + BETA, R2: ALPHA - BETA,
|
||||
RETURN(A1*%E^(R1*X) + A2*%E^(R2*X)))
|
||||
ELSE (BETA: SQRT(-1)*BETA,
|
||||
RETURN(%E^(ALPHA*X) * (A1*COS(BETA*X)
|
||||
+ A2*SIN(BETA*X)))))$
|
||||
DE: 'D(Y,X,2) - 'D(Y,X) - 6*Y = SIN(X);
|
||||
YH(X) := ''(SOLDER(%,Y,X));
|
||||
YP(X) := B1*SIN(X) + B2*COS(X)$
|
||||
YG(X) := YH(X) + YP(X)$
|
||||
PLUGIN: EV(DE,DIFF,EXPAND,Y=YP(X));
|
||||
EQN1: COEFF(PLUGIN,SIN(X));
|
||||
EQN2: COEFF(PLUGIN,COS(X));
|
||||
GLOBALSOLVE: TRUE$
|
||||
SOLN: LINSOLVE([EQN1,EQN2],[B1,B2]);
|
||||
YG(X);
|
||||
/* PLUGGING IN INITIAL CONDITIONS OF Y(0)=1
|
||||
AND Y'(0)=0 */
|
||||
|
||||
EQN1: YG(0) = 1;
|
||||
DIFF(YG(X),X);
|
||||
EQN2: EV(%,X=0) = 0;
|
||||
SOLN: LINSOLVE([EQN1,EQN2],[A1,A2]);
|
||||
YG(X);
|
||||
/* RESETTING OF OPTIONS */
|
||||
|
||||
GLOBALSOLVE: FALSE$
|
||||
"SOLUTION BY LAPLACE TRANSFORMS"$
|
||||
SUBST(Y(X),Y,DE);
|
||||
[ATVALUE(Y(X),X=0,1), ATVALUE('DIFF(Y(X),X),X=0,0)];
|
||||
LAPLACE(%TH(2),X,S);
|
||||
LINSOLVE([%],['LAPLACE(Y(X),X,S)]);
|
||||
ILT(%[1],S,X);
|
||||
BATCH(C2CYL,DEMO,DSK,DEMO);
|
||||
/* CONVERSION OF THE LAPLACIAN FROM CARTESIAN
|
||||
COORDS. TO CYLINDRICAL COORDS. */
|
||||
|
||||
/* CAUSE DERIVATIVES TO DISPLAY WITH SUBSCRIPTS */
|
||||
|
||||
DERIVABBREV:TRUE$
|
||||
/* ORDER X,Y, AND Z SO THEY WILL BE GROUPED NICELY */
|
||||
|
||||
ORDERLESS(Z,Y,X)$
|
||||
/* U(X,Y,Z) BECOMES U(R,T,Z) IN CYLINDRICAL COORDINATES
|
||||
R STANDS FOR RHO AND T FOR THETA */
|
||||
|
||||
DEPENDS(U,[R,T,Z])$
|
||||
/* INPUT THE TRANSFORMATION RULES FROM THE
|
||||
CARTESIAN SYSTEM TO THE CYLINDRICAL SYSTEM */
|
||||
|
||||
GRADEF(R,X,X/R)$
|
||||
GRADEF(R,Y,Y/R)$
|
||||
GRADEF(T,X,-Y/R^2)$
|
||||
GRADEF(T,Y,X/R^2)$
|
||||
/* SET EXPOP TO CAUSE PARENTHESIZED EXPRESSIONS
|
||||
TO BE EXPANDED AUTOMATICALLY */
|
||||
|
||||
EXPOP:1$
|
||||
/* NOW JUST INPUT THE LAPLACIAN IN CART. COORDS.,
|
||||
AND LET THE CHAIN RULE DO ITS THING */
|
||||
|
||||
DIFF(U,X,2)+DIFF(U,Y,2)+DIFF(U,Z,2);
|
||||
SUBST(R^2-X^2,Y^2,%);
|
||||
EQ:T^4*B(T)^3*DIFF(B(T),T,2)+(1-K*T^2)*B(T)^4-T^4;
|
||||
TRIAL:T+SUM(A[2*I+1]*T^(2*I+1),I,1,5);
|
||||
POWERDISP:TRUE$
|
||||
RATWEIGHT(T,1)$
|
||||
RATWTLVL:14$
|
||||
EV(EQ,B(T)=TRIAL,DIFF);
|
||||
EXPANDEDEQ:RAT(%,T);
|
||||
COEFF(EXPANDEDEQ,T,6);
|
||||
ANS3:SOLVE(%,A[3]);
|
||||
COEFF(EXPANDEDEQ,T,8);
|
||||
EV(%,ANS3);
|
||||
SOLVE(%,A[5]);
|
||||
/* ETC*/
|
||||
|
||||
FOR I:3 THRU 11 STEP 2 DO
|
||||
COEFFICIENT[I]:COEFF(EXPANDEDEQ,T,I+3)$
|
||||
FOR I:3 THRU 11 STEP 2 DO
|
||||
(SOL[I]:ANS:SOLVE(COEFFICIENT[I],A[I]),
|
||||
FOR J:I+2 STEP 2 THRU 11 DO
|
||||
COEFFICIENT[J]:EV(COEFFICIENT[J],ANS))$
|
||||
RATEXPAND:TRUE$
|
||||
FOR I:3 THRU 11 STEP 2
|
||||
DO PRINT(RATSIMP(EV(SOL[I])))$
|
||||
|
||||
Reference in New Issue
Block a user