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## Re: [Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivale

 From: William Sit Subject: Re: [Axiom-developer] 20080416.01.tpd.patch (CATS Schaums-Axiom equivalence testing (2-7)) Date: Thu, 17 Apr 2008 17:36:27 -0400

Tim:

You have a typo at bb:
bb:=1/(2*a^2*x*2)-1/(2*a^4)*log(x^2/(x^2-a^2))

x*2 should have been x^2; so this should be:

bb:=1/(2*a^2*x^2)-1/(2*a^4)*log(x^2/(x^2-a^2))

William

On Thu, 17 Apr 2008 06:26:26 -0400

address@hidden wrote:

Item 14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT in schaum7.input.pamphlet is particularly interesting because it appears that the derivative of Axiom's answer is the original integrand but the derivative of Schaum's answer is not, implying that Schaum's has a mistake. This will be verified using other systems later.


Axiom is weak in handling certain simplifications. Future work is
planned to correct this.


Richard Fateman has given me permission to use his TILU pattern integration database in Axiom. This should give us much broader integration results. TILU has not been tested against Schaums
but this testing will occur during the merge.

schaum2.input.pamphet

14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT

\section{\cite{1}:14.150~~~~~$\displaystyle\int{\frac{dx}{x^3(x^2-a^2)}}$}
@@ -124,7 +343,7 @@ 
<<*>>=
)clear all

---S 7 of 19
+--S 29
aa:=integrate(1/(x^3*(x^2-a^2)),x)

--R --R
@@ -135,6 +354,73 @@ aa:=integrate(1/(x^3*(x^2-a^2)),x)
--R                     2a x

--R Type: Union(Expression Integer,...) --E +
+--S 30
+bb:=1/(2*a^2*x*2)-1/(2*a^4)*log(x^2/(x^2-a^2))
+--R
+--R                     2
+--R                    x        2
+--R        - 2x log(-------) + a
+--R                  2    2
+--R                 x  - a
+--R   (2)  ----------------------
+--R                   4
+--R                 4a x

+--R Type: Expression Integer
+--E
+
+--S 31
+cc:=aa-bb
+--R
+--R                                                 2

+--R 2 2 2 2 2 x 2 2 +--R 2x log(x - a ) - 4x log(x) + 2x log(-------) - a x + 2a
+--R                                              2    2
+--R                                             x  - a

+--R (3) ---------------------------------------------------------
+--R                                    4 2
+--R                                  4a x

+--R Type: Expression Integer
+--E
+
+--S 32
+divlog:=rule(log(a/b) == log(a) - log(b))
+--R
+--R            a
+--R   (4)  log(-) == - log(b) + log(a)
+--R            b

+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+
+--S 33
+dd:=divlog cc
+--R
+--R          2     2      2          2      2
+--R        2x log(x ) - 4x log(x) - a x + 2a
+--R   (5)  ----------------------------------
+--R                         4 2
+--R                       4a x

+--R Type: Expression Integer
+--E
+
+--S 34
+logpow:=rule(log(a^n) == n*log(a))
+--R
+--R             n
+--R   (6)  log(a ) == n log(a)

+--R Type: RewriteRule(Integer,Integer,Expression Integer)
+--E
+

+--S 35 14:150 Schaums and Axiom DISAGREE BY A NON-CONSTANT
+ee:=logpow dd
+--R
+--R        - x + 2
+--R   (7)  -------
+--R           2 2
+--R         4a x

+--R Type: Expression Integer
+--E
+
@




I tried it on maximum and got:

(%i6) integrate(1/(x^3*(x^2-a^2)),x);
2      2

log(x - a ) log(x) 1 (%o6) ------------ - ------ + ------- 4 4 2 2 2 a a 2a x

and this agrees with my schaums book;

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William Sit, Professor of Mathematics, City College of New York Office: R6/202C Tel: 212-650-5179, Fax: 212-862-0004
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/