Simplification question
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From: Andrzej Kozlowski on 16 Jun 2010 05:38 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.
From: Patrick Scheibe on 16 Jun 2010 05:39 Hi, and Simplify[TrigToExp[ 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] == ArcTanh[Tanh[a] Tanh[b]]], Element[{a, b}, Reals]] doesn't help? Cheers Patrick On Tue, 2010-06-15 at 02:30 -0400, 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? >
From: Bob Hanlon on 16 Jun 2010 05:40 $Version 7.0 for Mac OS X x86 (64-bit) (February 19, 2009) eqn = 1/2 Log[(Exp[a + b] + Exp[-a - b])/ (Exp[-a + b] + Exp[a - b])] == ArcTanh[Tanh[a] Tanh[b]]; eqn // TrigToExp // Simplify[#, Element[{a, b}, Reals]] & True eqn // TrigToExp // Simplify // PowerExpand True Bob Hanlon ---- Yaroslav Bulatov <yaroslavvb(a)gmail.com> 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?
From: Simon on 16 Jun 2010 05:40 It might be that Mathematica is having troubles because of the branch cuts in the inverse functions. If you take Tanh of both sides, then it becomes easier... On Jun 15, 4:30 pm, Yaroslav Bulatov <yarosla...(a)gmail.com> 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?
From: Alexei Boulbitch on 16 Jun 2010 05:41 FullSimplify[( 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])])/ ArcTanh[Tanh[a] Tanh[b]] , Assumptions -> {a > 0 && b > 0}] 1 FullSimplify[ 1/2 Log[(Exp[a + b] + Exp[-a - b])/(Exp[-a + b] + Exp[a - b])] - ArcTanh[Tanh[a] Tanh[b]] , Assumptions -> {a > 0 && b > 0}] 0 ?? 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? -- Alexei Boulbitch, Dr. habil. Senior Scientist Material Development IEE S.A. ZAE Weiergewan 11, rue Edmond Reuter L-5326 CONTERN Luxembourg Tel: +352 2454 2566 Fax: +352 2454 3566 Mobile: +49 (0) 151 52 40 66 44 e-mail: alexei.boulbitch(a)iee.lu www.iee.lu -- This e-mail may contain trade secrets or privileged, undisclosed or otherwise confidential information. If you are not the intended recipient and have received this e-mail in error, you are hereby notified that any review, copying or distribution of it is strictly prohibited. Please inform us immediately and destroy the original transmittal from your system. Thank you for your co-operation.
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