[EMAIL PROTECTED] wrote:
On at least 2 of these problems Schaums and Axiom seem to disagree
on the results, namely:
14:569 SCHAUMS AND AXIOM DISAGREE?
14:571 SCHAUMS AND AXIOM DISAGREE?
as the compute result is not constant but does not seem to have a
simplification that eliminates x.
This needs to be studied further.
---S 8 of 28
+--S 17
aa:=integrate(cosh(a*x)^2,x)
--R
--R
@@ -170,6 +248,42 @@ aa:=integrate(cosh(a*x)^2,x)
--R 2a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 18
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2
this should be
+bb:=x/2+(sinh(a*x)*cosh(a*x))/2a
I think your Schaums is out of date on this one.
+--R
+--R cosh(a x)sinh(a x) + x
+--R (2) ----------------------
+--R 2
+--R Type: Expression
Integer
+--E
+
+--S 19
+cc:=aa-bb
+--R
+--R (- a + 1)cosh(a x)sinh(a x)
+--R (3) ---------------------------
+--R 2a
+--R Type: Expression
Integer
+--E
+
+--S 20
+sinhcoshrule:=rule(sinh(x)*cosh(y) == 1/2*(sinh(x+y)+sinh(x-y)))
+--R
+--I %B sinh(y + x) - %B sinh(y - x)
+--I (4) %B cosh(y)sinh(x) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 21 14:569 SCHAUMS AND AXIOM DISAGREE?
+dd:=sinhcoshrule cc
+--R
+--R (- a + 1)sinh(2a x)
+--R (5) -------------------
+--R 4a
+--R Type: Expression
Integer
+--E
@
For this next one Maxima gives
(%i6) integrate(1/cosh(a*x)^2,x);
(%o6) -4/(2*a*%e^(2*a*x)+2*a)
But if I try
(%i7) integrate(1/cos(a*x*i)^2,x);
(%o7)
(2*sin(2*a*i*x))/(a*i*sin(2*a*i*x)^2+a*i*cos(2*a*i*x)^2+2*a*i*cos(2*a*i*x)+a*i)
(%i8) trigreduce(%);
(%o8) sin(2*a*i*x)/(a*i*cos(2*a*i*x)+a*i)
(%i9) trigrat(%);
(%o9) sin(a*i*x)/(a*i*cos(a*i*x))
(%i10) trigreduce(%);
(%o10) tan(a*i*x)/(a*i)
Now this can be written as
tanh(ax) /ai
and this is different than Schaums
I think my logic is OK here But I am not an expert.
\section{\cite{1}:14.571~~~~~$\displaystyle
@@ -201,7 +396,7 @@ $$
<<*>>=
)clear all
---S 10 of 28
+--S 31
aa:=integrate(1/cosh(a*x)^2,x)
--R
--R
@@ -211,6 +406,176 @@ aa:=integrate(1/cosh(a*x)^2,x)
--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
--R Type: Union(Expression
Integer,...)
--E
+
+--S 32
+bb:=tanh(a*x)/a
+--R
+--R tanh(a x)
+--R (2) ---------
+--R a
+--R Type: Expression
Integer
+--E
+
+--S 33
+cc:=aa-bb
+--R
+--R 2 2
+--R (- sinh(a x) - 2cosh(a x)sinh(a x) - cosh(a x) - 1)tanh(a x) - 2
+--R (3) ------------------------------------------------------------------
+--R 2 2
+--R a sinh(a x) + 2a cosh(a x)sinh(a x) + a cosh(a x) + a
+--R Type: Expression
Integer
+--E
+
+--S 34
+tanhrule:=rule(tanh(x) == sinh(x)/cosh(x))
+--R
+--R sinh(x)
+--R (4) tanh(x) == -------
+--R cosh(x)
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 35
+dd:=tanhrule cc
+--R
+--R (5)
+--R 3 2 2
+--R - sinh(a x) - 2cosh(a x)sinh(a x) + (- cosh(a x) - 1)sinh(a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 2 3
+--R a cosh(a x)sinh(a x) + 2a cosh(a x) sinh(a x) + a cosh(a x) + a
cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 36
+sinhsqrrule:=rule(sinh(x)^2 == 1/2*cosh(2*x)-1/2)
+--R
+--R 2 cosh(2x) - 1
+--R (6) sinh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 37
+ee:=sinhsqrrule dd
+--R
+--R (7)
+--R 3 2
+--R - 2sinh(a x) + (- 2cosh(a x) - 2)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 2 3
+--R 4a cosh(a x) sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 38
+coshsqrrule:=rule(cosh(x)^2 == 1/2*cosh(2*x)+1/2)
+--R
+--R 2 cosh(2x) + 1
+--R (8) cosh(x) == ------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 39
+ff:=coshsqrrule ee
+--R
+--R (9)
+--R 3
+--R - 2sinh(a x) + (- cosh(2a x) - 3)sinh(a x) - 2cosh(a x)cosh(2a x)
+--R +
+--R - 2cosh(a x)
+--R /
+--R 3
+--R (2a cosh(2a x) + 2a)sinh(a x) + a cosh(a x)cosh(2a x) + 2a cosh(a x)
+--R +
+--R a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 40
+sinhcuberule:=rule(sinh(x)^3 == 1/4*sinh(3*x)-3/4*sinh(x))
+--R
+--R 3 sinh(3x) - 3sinh(x)
+--R (10) sinh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 41
+gg:=sinhcuberule ff
+--R
+--R (11)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
+--R +
+--R - 4cosh(a x)
+--R /
+--R
3
+--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(a x)cosh(2a x) + 4a cosh(a x)
+--R +
+--R 2a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 42
+coshcuberule:=rule(cosh(x)^3 == 1/4*cosh(3*x)-3/4*cosh(x))
+--R
+--R 3 cosh(3x) - 3cosh(x)
+--R (12) cosh(x) == -------------------
+--R 4
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 43
+hh:=coshcuberule gg
+--R
+--R (13)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 4cosh(a x)cosh(2a x)
+--R +
+--R - 4cosh(a x)
+--R /
+--R (4a cosh(2a x) + 4a)sinh(a x) + a cosh(3a x) + 2a cosh(a x)cosh(2a x)
+--R +
+--R - a cosh(a x)
+--R Type: Expression
Integer
+--E
+
+--S 44
+coshcoshrule:=rule(cosh(x)*cosh(y)==1/2*(cosh(x+y)+cosh(x-y)))
+--R
+--I %U cosh(y + x) + %U cosh(y - x)
+--I (14) %U cosh(x)cosh(y) == -------------------------------
+--R 2
+--R Type: RewriteRule(Integer,Integer,Expression
Integer)
+--E
+
+--S 45
+ii:=coshcoshrule hh
+--R
+--R (15)
+--R - sinh(3a x) + (- 2cosh(2a x) - 3)sinh(a x) - 2cosh(3a x) - 6cosh(a x)
+--R ----------------------------------------------------------------------
+--R (4a cosh(2a x) + 4a)sinh(a x) + 2a cosh(3a x)
+--R Type: Expression
Integer
+--E
+
+--S 46 14:571 SCHAUMS AND AXIOM DISAGREE?
+jj:=complexNormalize ii
+--R
+--R a x 2
+--R - 2(%e ) - 2
+--R (16) ---------------
+--R a x 2
+--R 2a (%e ) - a
+--R Type: Expression
Integer
+--E
_______________________________________________
Axiom-developer mailing list
Axiom-developer@nongnu.org
http://lists.nongnu.org/mailman/listinfo/axiom-developer