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.