Re: RE: Re: Showing that ArcSinh[2]/ArcCsch[2] is 3?

*To*: mathgroup at smc.vnet.net*Subject*: [mg73685] Re: [mg73623] RE: [mg73550] Re: Showing that ArcSinh[2]/ArcCsch[2] is 3?*From*: János <janos.lobb at yale.edu>*Date*: Sat, 24 Feb 2007 02:25:47 -0500 (EST)*References*: <200702230936.EAA17628@smc.vnet.net>

Interestingly the front end can be quite misleading to the eye :) In[18]:= N[ArcSinh[2]/ArcCsch[2], {Infinity, 423}] /423 is the maximum achievable numerical accuracy I have on my G5/ In the notebook I see: 3.0000000000000000000000000000000000000000000000000000000000000000000000= 000000\ 000000000000000000000000000000000000000000000000000000000000000000000000= 000000\ 000000000000000000000000000000000000000000000000000000000000000000000000= 000000\ 000000000000000000000000000000000000000000000000000000000000000000000000= 000000\ 000000000000000000000000000000000000000000000000000000000000000000000000= 000000\ 00000000000000000000000000000000000 However, In[23]:= IntegerPart[N[ArcSinh[2]/ ArcCsch[2], {Infinity, 423}]] Out[23]= 2 Hehe...haha... This is the situation when just by looking in the front end, the integer part of a Real 3. becomes 2 J=E1nos P.S. By the way if I increase my accuracy to 424 in N[], the local kernel quits :) On Feb 23, 2007, at 4:36 AM, Tony Harker wrote: > Dear Oleksandr, > > That's interesting: but looking at what is produced by > ArcSinh[2]/ArcCsch[2] == 3 // FullSimplify > makes me wonder just how Mathematica achieved this. In other > words, what > does the message that Bob Hanlon so carefully suppressed mean? Similar > messages sometimes occur when checking the results of simple > equations by > back-substituting, especially when the solutions contain surds, and = > appear > to show that Mathematica is diving off into real numbers to prove > results > involving integers and powers of integers. > > Tony > > > Dr A.H. Harker > Department of Physics and Astronomy > University College London > Gower Street > London > WC1E 6BT > > > > ]->-----Original Message----- > ]->From: sashap [mailto:pavlyk at gmail.com] > ]->Sent: 21 February 2007 11:04 > ]->To: mathgroup at smc.vnet.net > ]->Subject: [mg73550] Re: Showing that ArcSinh[2]/ArcCsch[2] is 3? > ]-> > ]->Dear David, > ]-> > ]->It is the case in the CAS and in the real life as well that > ]->proving left hand side equal to right hand side is easier > ]->than deriving what right hand side the left hand side equals to. > ]-> > ]->I can not think of a direct way to use built-in 'knowledge' > ]->to reduce ArcSinh[2]/ArcCsch[2] to 3, but want to point out > ]->that proving it post-factum does not seem problematic: > ]-> > ]->In[3]:= > ]->ArcSinh[2]/ArcCsch[2]-3//FullSimplify > ]-> > ]->Out[3]= > ]->0 > ]-> > ]->Sincerely, > ]-> > ]->Oleksandr Pavlyk > ]->Special Functions Developer > ]->Wolfram Research Inc > ]-> > ]->On Feb 20, 5:24 am, "David W.Cantrell" > ]-><DWCantr... at sigmaxi.net> wrote: > ]->> I hope I've just overlooked something very simple. > ]->> I want to transform ArcSinh[2]/ArcCsch[2] to 3, using just > ]->"knowledge" > ]->> already implemented in Mathematica. I tried FullSimplify > ]->first, and it > ]->> doesn't help. I tried several other things. For example, > ]->> > ]->> TrigToExp[ArcSinh[2]/ArcCsch[2]] yields > ]->> > ]->> Log[2 + Sqrt[5]]/Log[1/2 + Sqrt[5]/2]. > ]->> > ]->> But then how should we transform that to 3? > ]->> > ]->> David > ]-> > ]-> > ]-> > ]-> >

**References**:**RE: Re: Showing that ArcSinh[2]/ArcCsch[2] is 3?***From:*"Tony Harker" <a.harker@ucl.ac.uk>