Re: Re: Re: Hyperbolic function identity

• To: mathgroup at smc.vnet.net
• Subject: [mg50996] Re: [mg50964] Re: [mg50945] Re: [mg50932] Hyperbolic function identity
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Fri, 1 Oct 2004 04:48:12 -0400 (EDT)
• References: <200409300852.EAA26465@smc.vnet.net> <EC6856EA-12D4-11D9-8846-000A95B4967A@mimuw.edu.pl>
• Sender: owner-wri-mathgroup at wolfram.com

```Here is a puzzling observation.

FullSimplify[FullSimplify[ArcCosh[1 + z^2/2] -
2*ArcSinh[z/2], ComplexityFunction ->
(1/(1 + Count[#1, Log, Infinity, Heads ->
True]) & )], z > 0]

0

I used a silly looking ComplexityFunction whose only purpse was to
force FullSimplify to choose expressions involving logs in preference
to those without them. FullSimplify obviously made use of TrigToExp or
something like that to get the answer. However, the really curious
thing is this:

FullSimplify[ArcCosh[1 + z^2/2] - 2*ArcSinh[z/2], z > 0,
ComplexityFunction ->
(1/(1 + Count[#1, Log, Infinity, Heads -> True]) & )]

2*(Log[2] - Log[z + Sqrt[z^2 + 4]]) +
Log[z^2 + Sqrt[z^2 + 4]*z + 2] - Log[2]

FullSimplify[%, z > 0]

0

The fact that two applications of FullSimplify were needed seems
strange to me and it might be a bug. One would have exected that having
obtained the first expression above FullSimplify would make one more
attempt at  FullSImplifying thus obtaining the answer 0. Somehow it
fails to do so.

Andrzej

On 30 Sep 2004, at 20:36, Andrzej Kozlowski wrote:

> It seems to me that the problem lies elsewhere. Note that
>
> FullSimplify[ArcCosh[1 + z^2/2] - 2*ArcSinh[z/2], z > 0,
>    TransformationFunctions -> {Automatic, TrigToExp}]
>
> does not work and you have to use:
>
> Simplify[ArcCosh[1+z^2/2]-2*ArcSinh[z/
>          2],z>0,TransformationFunctions->{Automatic,
>      FullSimplify[TrigToExp[#],z>0]&}]
>
> It looks to me more likely that TrigToExp is actually used as a
> transformation function by FullSimplify while searching for the
> "simplest" form, but unless the condition z>0 is also used at the same
> time the expressions obtained in this way do not actually become
> "simpler" and are discarded. I don't think this can be helped unless
> FullSimplify with the condition z>0 is itself included among the
> transformation functions (which of course seems rather pointless in
> this case as you can just apply it to the entire expression itself).
>
> Andrzej
>
> On 30 Sep 2004, at 17:52, Wolf, Hartmut wrote:
>
>>
>> No course to blame Mathematica. We do have TrigToExp and ExpToTrig,
>> but
>> cannot keep them both at the same time (no confluent ruleset).  So
>> without
>> meta-rules, a choice must be made by the user.
>>
>> --
>> Hartmut
>>
>>
>>> -----Original Message-----
>>> From: Maxim A. Dubinnyi [mailto:maxim at nmr.ru]
To: mathgroup at smc.vnet.net
>>> Sent: Wednesday, September 29, 2004 9:15 AM
>>> To: mathgroup at smc.vnet.net
>>> Subject: [mg50996] [mg50964] [mg50945] Re: [mg50932] Hyperbolic function
>>> identity
>>>
>>>
>>> This works correctly:
>>>
>>> FullSimplify[TrigToExp[ArcCosh[1+(z^2)/2]-2*ArcSinh[z/2]], z > 0]
>>>
>>> It looks like that mathematica works better with logarithms and
>>> exponents then with trigonometric functions.
>>>
>>> Carlos Felippa wrote:
>>>
>>>> Why
>>>>
>>>>   FullSimplify[ArcCosh[1+z^2/2]-2*ArcSinh[z/2],z>0];
>>>>
>>>> does not evaluate to 0?
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>
>

```

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