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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