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From: Yaroslav Bulatov on 15 Jun 2010 02:30
I'd like to verify that the following expression is true for a,b real.
It seems to hold numerically.

1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
ArcTanh[Tanh[a] Tanh[b]]

I tried Reduce and combinations of TrigToExp/Simplify with no luck,
any suggestions?

From: Bill Rowe on 16 Jun 2010 05:36
On 6/15/10 at 2:30 AM, yaroslavvb(a)gmail.com (Yaroslav Bulatov) wrote:

>I'd like to verify that the following expression is true for a,b
>real. It seems to hold numerically.

>1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
>ArcTanh[Tanh[a] Tanh[b]]

>I tried Reduce and combinations of TrigToExp/Simplify with no luck,
>any suggestions?

In[7]:= eq =
1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
ArcTanh[Tanh[a] Tanh[b]];

In[8]:= Assuming[{a, b} \[Element] Reals, FullSimplify[TrigToExp[eq]]]

Out[8]= True

In[9]:= $Version

Out[9]= 7.0 for Mac OS X x86 (64-bit) (February 19, 2009)


From: Peter Breitfeld on 16 Jun 2010 05:37
Yaroslav Bulatov wrote:

> I'd like to verify that the following expression is true for a,b real.
> It seems to hold numerically.
>
> 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
> ArcTanh[Tanh[a] Tanh[b]]
>
> I tried Reduce and combinations of TrigToExp/Simplify with no luck,
> any suggestions?
>

At least for Reals you can verify it:

1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
ArcTanh[Tanh[a] Tanh[b]] // TrigToExp //
FullSimplify[#, {a, b} \[Element] Reals] &


//Peter
--
_________________________________________________________________
Peter Breitfeld, Bad Saulgau, Germany -- http://www.pBreitfeld.de

From: Daniel Huber on 16 Jun 2010 05:37
Hi,
sometimes mathematica needds just a little help from a friend...
Assuming[Element[{a, b,}, Reals],
1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
ArcTanh[Tanh[a] Tanh[b]] // TrigToExp // Simplify]

cheers, Daniel

Am 15.06.2010 08:30, schrieb Yaroslav Bulatov:
> I'd like to verify that the following expression is true for a,b real.
> It seems to hold numerically.
>
> 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ==
> ArcTanh[Tanh[a] Tanh[b]]
>
> I tried Reduce and combinations of TrigToExp/Simplify with no luck,
> any suggestions?
>


From: Adriano Pascoletti on 16 Jun 2010 05:37
Considering the difference of the two sides the answer is returned
instantly


In[1]:== FullSimplify[1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a
- b])] -
ArcTanh[Tanh[a] Tanh[b]], a =E2=88=88 Reals && b =E2=88=88 Reals]
Out[1]== 0


Adriano Pascoletti

2010/6/15 Yaroslav Bulatov <yaroslavvb(a)gmail.com>

> I'd like to verify that the following expression is true for a,b real.
> It seems to hold numerically.
>
> 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] ====
> ArcTanh[Tanh[a] Tanh[b]]
>
> I tried Reduce and combinations of TrigToExp/Simplify with no luck,
> any suggestions?
>
>


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