. , , ,

,,,

Maple — ,

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" Maple "

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, 2008


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1.

Maple , , , .

:

(_1, _2,, _n);

, _1, _2, , , . , , , , , .

Maple . , ( , ). , , simplify () , .

. ( ) , . , , . . , , value () . . (diff Diff), (int Int) .

1. .

> k:=Int (cos(x)^3, x);

 

> k=int (cos(x)^3, x);

 

> value(k);

 

Maple, . , . readlib () with (). , . , .

Maple, , , . , , , .. , .

2. : simplify ()

simplify () , , . , Maple , , , , .

. : simplify ().

, . simplify () , , . simplify () , Maple. , (, <F1>): `simplify/exp` , `simlif/ln` , `simplif/sqrt` , , `simplif/trig` , `simplif/radical` ( ), `simplif/power` , .. Maple , .

, , , . : simlif (, nl, n2,);

nl, n2 .. : Ei, GAM, RootOf, @, hypergeom, ln, polar, power, radical, sqrt, trig. ? simplify [], [] .

, , . assume=. : simplify (, ssum=); : complex , real , positive , integer , RealRange (a, b) (, b) .

simplify() :

2. .

> c:=ln (exp(x))+x*ln (exp(x));

> simplify(c);

> simplify (c, assume=real);


> d:=1/sqrt(8)*(((1+sqrt(8))/10)^5+((1‑sqrt(8))/10)^5);

> simplify(d);

2, ln (exp(x))+x·ln (exp(x)), . Maple (.. , ). .

3. .

> f:=(sqrt (x^2));

> simplify(f);

> simplify (f, assume=real);

> simplify (f, assume=positive);

simplify() . , :

{l, 2,}

- , , :

> k:=a+b^2+c^3+d+5;

> simplify (k, {c^3+d, a+b^2=1});

, c^3+d 0.

, , , . : . ( , , Maple, , .) , , , , . , , ..

4. .

> equ:={sin(x)^2+cos(x)^2=1};

e:=sin(x)^311*sin(x)^2*cos(x)+3*cos(x)^3‑sin(x)*cos(x)+2;

> simplify (e, equ, [sin(x), cos(x)]);

> simplify (e, equ, [cos(x), sin(x)]);

> simplify (e, equ, {sin(x), cos(x)});

> simplify (e, equ, {cos(x), sin(x)});

3. : expand ()

expand () , .. . . ( ) , .

: nd (, l, 2,, n); , , l, 2, n , .

5. .

> expand((x+3)*(x+4)^2);

 

> expand((x+3)^3/(x+4)^2);

 

> expand (cos(x-y));

 

> expand((x+3)*(x+4)^2, x+3);

 

> expand (x^((a+b)*(k+f)));

 


4. : factor ()

factor () . Maple , . , .. , , , :

> factor (x^3*y‑2*x^2*a*y+x*y*a^2‑x^3*b^2+2*x^2*b^2*a-x*b^2*a^2+x^2*y^22*x*y^2*a+y^2*a^2‑y*b^2*x^2+2*y*b^2*x*a-y*b^2*a^2);

 

: , . , . , . real, complex,  / . 6 .

6. .

> factor (x^3+2); # ( )

> factor (x^3+2.0); #

( )

> factor (x^3+2, real); #

( real)

> factor (x^3+2, complex); #

( complex)

> factor (x^3+2,2^(1/3)); # 2^(1/3)

( )

factor () ( ), ( ), , ( ):

> d:=(x^11‑y^11)/(x^6‑y^6);

 


> factor(d);

 

5. : normal ()

normal () , , , , . : normal (f); normal (f, expanded); f , expanded , .

7. .

> f:=1/x+1/(x+1)^2+1/(x+1);

> normal(f);

f , , , , , , normal () f. , , , . , , , , , :

> s:=sin (x/(x+1)  x)^2+cos (-x/(x+1)+x);

> normal(s);

> normal (1/x+y=x/y+(3*y)/x);

6. : combine ()

combine () , , , , . , , , expand (). , : sin (+b) = sin(a) cos(b) +cos() sin(b).

expand () , combine () :

> expand (sin(a+b));


> combine (sin(a)*cos(b)+cos(a)*sin(b));

:

> g:=sin (a+b)^2;

> s:=expand(g);

> f:=combine(s);

, combine () s g, expand (). , l , , ( ) combine (). , . g, subs (), , :

> subs (cos(2*a+2*b)=-2*sin (a+b)^2+1, f);


combine () . :

abs exp piecewise Psi Signum

arctan icombine polylog radical trig

conjugate ln power range

Maple , . , (arctan) (ln, radical), symbolic, combine () , .

7. : collect ()

collect () , , Maple. :

collect (, );

llt (, , form, func);

llt (, x, func);

, . - , .

collect () , , .. .

8. .

k:=x^3*sin(x)^2+x^3*cos(x)+x^3*exp(x)+x*cos(x)+2*x*exp(x)+7*x*sin(x)+4*x^3;

 

> collect (k, x);

 

> collect (k, x^3);

 

> collect (k, exp(x));

 

> collect (k, sin(x));

 

> collect (k, cos(x));

 


8 .

form . , , , . form : recursive distributed. , .. form , , , Maple . distributed , , .

9. .

> p:=x*y-a^2*x*y+y*x^2‑a*y*x^2+x+a*x; #

> collect (p, [x, y], recursive);

> collect (p, [y, x], recursive);


> collect (p, {x, y}, recursive);

> collect (p, {x, y}, distributed);

> collect (p, [x, y], distributed);

func , . simplify () factor ().

10. , .

> d:=a^4*y-y+a^4+a^2;

 

> collect (d, y);

 

> collect (d, y, factor); # y

 


8. : rationalize ()

. rationalize () . . , sin (), (), ln () .. , .

11. .

> a:=7*(3^(1/3)+4^(1/5))/(32^(1/3));

 

> rationalize(a);

 

> b:=y/(y+sqrt (2‑sqrt(5)));

 

> rationalize(b);

 


> c:=1/(3‑root (cos(1/(2+sqrt(mu))), 5));

 

> rationalize(c);

 

 

9. : assume ()

, , .. - . , , . l , . Maple, simplify(), sqrt(), , .

assume () Maple. :

assume (x, );

, , ( , Maple , ), . . 1.

1.

negative

(-¥, 0)

( )

nonnegative

(0,¥)

( )

positive

(0,¥)

( )

natural (, 0)
posint

0

odd

even

complex

NumeralNonZero

, 0

real

rational ( )
irrational

integer

fraction

prime

(, ) , , , . , (x, negative) <0, (, nonnegative) >=0 ..

- , , (~). :

¨ , , ( Options Þ Assumed Variables Þ No Annotation);

¨ , , , , ( Options Þ Assumed Variables Þ Phrase).

12. .

> assume (a>0);

> ln (a^2); #

> ln (a^2); #

> ln (a^2); #

Options Þ Assumed Variables Þ Trailing Tildes.

assume () (, ) . .

> ssum (>3, <5);

, (3,5).

, assume () , . :

> assume (x>3);

> assume (x<5);

, 5, , (3,5).

, additionally(), assume (). , additionally (), , assume () additionally ():

> assume (x>3); # >3

(- )

> dditinll (<=5); # , 3<<=5

( , ). , :

> x:='x';

, . 13.

13. .

> assume (b>0);

> d:=surd (b^4,4);

 

> b:='b':b;

> d;

, b , , b d. , subs () b 'b'.

14. .

> assume (b>0);

> d:=sqrt (b^4);

 

> d;

 

> d:=subs (b='b', d);

 

> b:='b';

 


> d;

 

is () , . true, . , is () false. is () FAIL, , . , .

15. .

> assume (a>0);

> is (a>0);

> is (a<1);

> additionally (a<1);

> is (a<1);

coulditbe () , . true, , f1s . FAIL is().

16. .

> assume (a>0);

> is (a>0);

> coulditbe (a=1);

> additionally (a<1);

> coulditbe (a=1);

about () :

> about(a);

Originally a, renamed a~:

is assumed to be: RealRange (Open(0), Open(1))

, Maple . , le - :

> int (exp(a*x), x=0..infinity);

Definite integration: Can't determine if the integral is convergent.

Need to know the sign of > a

Will now try indefinite integration and then take limits.

, a>0, Maple , , :

> assume (a>0);

> int (exp(a*x), x=0..infinity);


1.  ..,  ..  Maple. . .: , 1997. 208 .

2.  ..  Maple V. .: , 1998.

3.  ..  . .: . - , 1983. 176 .

4.  .. Maple 6. . .: , 2001.  528 .

5.  .. Maple V Power Edition .: - , 1998 .

: : &quot; Maple &quot;

 

 

 

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