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Re: One fundamental Gudermannian identity not verified

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91994] Re: One fundamental Gudermannian identity not verified
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Tue, 16 Sep 2008 19:24:13 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <gal3k0$e1n$1@smc.vnet.net>

sigismond kmiecik wrote:

> Since I couldn't  find a definition of the Gudermannian fonction among
> Mathematica packages I defined it myself :
> 
>> In[9]:=       Gd= (2 tan^-1(e^#)-\[Pi]/2) &
========================????????
The above expression *cannot* yield the result below for neither 
lowercase e has any built-in meaning nor tan^-1. However, capital E and 
ArcTan have the desired meaning.

>>
>> Out[9]=      2 ArcTan[\[ExponentialE]^#1] - \[Pi]/2 &
> 
> I could verify all the identies with that  fonction I thought of ,
> including one involving half-angles ie
> 
>> In[14]:= FullSimplify[Subtract @@ {#[[1]][Gd[x]/2], #[[2]][x/2]}, 
>>    x\[Epsilon] Reals] & /@ {{Tan, Tanh}}
========??????????
Surely you meant Element[x, Reals] and not x times epsilon times Reals?

>>
>> Out[14]= {0}
> 
> but not this last one :
> 
>> In[16]:=      FullSimplify[Subtract @@ {#[[1]][x], #[[2]][Tanh x/2]}, 
=============================================================?????????
What you have written means (Tanh times x) divided by two.

>>                        x\[Epsilon] Reals] & /@ {{Gd , 2 ArcTan}}
>>
>> Out[16]=    {-\[Pi]/2 + 
>>                         2 ArcTan[\[ExponentialE]^x] - (2 ArcTan)[(Tanh x)/2]}
=========================================================????????????
Notice the spurious set of parentheses around two times ArcTan...

> Is there another way with Mahematica 6.0 to make this verification?

You should post actual code, i.e. the output must be produce by the 
input, and not a mix of fantasied notation.

Regards,
-- Jean-Marc


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