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## Re[2]: [Axiom-developer] [Q] How to classify # integrate(log(%i+z^2), z)

 From: Vladimir Bondarenko Subject: Re[2]: [Axiom-developer] [Q] How to classify # integrate(log(%i+z^2), z) -> 2 ? Date: Mon, 14 Feb 2005 18:15:28 +0200

```Hi,

Monday, February 14, 2005, 4:16:03 PM, you write:

WS> Axiom may (I am not sure) have interpreted %i as a parameter
WS> with no special property.

If it is the case, this is certainly a bug as %i has a special,
a unique meaning in AXIOM, isn't is so?

-> #integrate(log(%i+z^2), z)
2

mean that in fact the user would have 2 identical answers,
and if we ask AXIOM to calculate

-> 2*3

we sure do not expect to see

[6,6]

but rather

6

?

Even more, if these 2 answers returned by AXIOM for
integrate(log(%i+z^2), z) are NOT identical, it is
obviously a math bug. The must be the same (but maybe
of different form).

Best,

WS> Bill Page wrote:
>>
>> On Friday, February 11, 2005 9:56 AM Vladimir Bondarenko wrote:
>> > ...
>> > ===> Case 2.
>> >
>> > -> integrate(log(%i+z^2), z)
>> >
>> >                   +-----+    2
>> >      +-----+    z\|- 4%i  + z  - %i            2
>> >     \|- 4%i log(-------------------) + 2z log(z  + %i) - 4z
>> >                        2
>> >                       z  + %i
>> >    [-------------------------------------------------------,
>> >                                2
>> >                                  +---+
>> >            2          +---+     \|4%i
>> >     z log(z  + %i) - \|4%i atan(------) - 2z]
>> >                                   2z
>> > ...
>> > But why the user really needs 2 forms of the same answer?
>> >
>>
>> I think you should report it as a bug.
>>
WS> I'm not sure, because you can use

(4) ->> complexIntegrate(log(%i+z^2),z)

WS>    (4)
WS>                     +-----+
WS>         +-----+    \|- 4%i  + 2z            2
WS>        \|- 4%i log(-------------) + 2z log(z  + %i)
WS>                          2
WS>      +
WS>                         +-----+
WS>           +-----+    - \|- 4%i  + 2z
WS>        - \|- 4%i log(---------------) - 4z
WS>                             2
WS>   /
WS>      2
WS>                             Type: Expression Complex Integer

WS> to obtain one form. Using integrate when the integrand involves complex
values
WS> and ask for real forms does not sound correct. Axiom may (I am not sure)
have
WS> interpreted %i as a parameter with no special property.

WS> A comparison with a parameter to replace %i shows the form of the answers in
WS> both cases (integrate or complexIntegrate) are the same.

(5) ->> complexIntegrate(log(a+z^2),z)

WS>    (5)
WS>                +----+                       +----+
WS>     +----+    \|- 4a  + 2z     +----+    - \|- 4a  + 2z            2
WS>    \|- 4a log(------------) - \|- 4a log(--------------) + 2z log(z  + a) -
4z
WS>                     2                           2
WS>
---------------------------------------------------------------------------
WS>                                         2
WS>                                                      Type: Expression
Integer
(6) ->> integrate(log(a+z^2),z)

WS>    (6)
WS>                  +---+    2
WS>      +---+    2z\|- a  + z  - a           2
WS>    [\|- a log(-----------------) + z log(z  + a) - 2z,
WS>                      2
WS>                     z  + a
WS>                  +-+
WS>         +-+     \|a            2
WS>     - 2\|a atan(----) + z log(z  + a) - 2z]
WS>                   z
WS>                                      Type: Union(List Expression
Integer,...)

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```