I just tried what happens using the definition
myacos z == 4*atan(sqrt((1-z)/2)/(1+sqrt((1+z)/2)))
In the testsuite I found the integral
in211:=integrate(acos(sin(2*z))*cos(z), z= 0..4*%pi/3)
--
-- +-+
-- 13%pi\|3 + 36
-- --------------
-- 12
-- Type: Union(f1: OrderedCompletion Expression Integer,...)
Well, mathematica6 gives
In[1]:= Integrate[ArcCos[Sin[2*z]]*Cos[z], {z,0,4*Pi/3}]
Limit::ztest: Unable to decide whether numeric quantities
-I/3 Pi I Sqrt[3] (-2 I)/3 Pi I Sqrt[3]
{-E - --------- - E + -----------}
I/3 Pi (2 I)/3 Pi
E E
are equal to zero. Assuming they are.
Pi
Out[1]= 1 - 2 Sqrt[2] - ---------
4 Sqrt[3]
In[2]:= N[%]
Out[2]= -2.28188
instead. This agrees with
(63) -> romberg(z +-> myacos(sin(2*z))*cos(z), 0, 4*%pi/3, 0.1, 0.1,10,20)
(63)
[value= - 2.2818769658 047445609, error= 0.28 E -17, totalpts= 2049,
success= true]
Type: Record(value: Float,error: Float,totalpts: Integer,success: Boolean)
(Actually, it doesn't matter whether I take myacos or acos here, since we are
on the real line...)
The reason that in211 differs is simply
(68) -> acos(sin(2*z))*cos(z)
(- 4z + %pi)cos(z)
(68) ------------------
2
Type: Expression(Integer)
which seems to be something different. However:
(65) -> int := integrate(myacos(sin(2*z))*cos(z), z= 0..4*%pi/3, "noPole")
+-+ +-+ +-+
+-+ \|2 \|3 - \|2 + 1
(65) 2\|3 atan(---------------------------) + 3
+-+ +-+ +-+
(2\|2 + 1)\|3 + 2\|2 + 6
Type: Union(f1: OrderedCompletion(Expression(Integer)),...)
(66) -> numeric int
(66) 3.4534498410 585544626
:-( Not sure how to go on... How could I find out what transformations FriCAS
:is using here?
Martin
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