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Re: Simplification question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg110389] Re: Simplification question
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Wed, 16 Jun 2010 05:39:00 -0400 (EDT)
On 15 Jun 2010, at 15:30, 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?
>
Fix b and note that you get an analytic function of one variable a. Now note that
D[(1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] -
ArcTanh[Tanh[a] Tanh[b]]), a] // Simplify
0
and
1/2 ( Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] -
ArcTanh[Tanh[a] Tanh[b]]) /. a->0
0
Since the function is analytic, this is enough to prove that the expression is identically zero.
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