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One fundamental Gudermannian identity not verified

  • To: mathgroup at smc.vnet.net
  • Subject: [mg91968] One fundamental Gudermannian identity not verified
  • From: sigismond kmiecik <sigismond.kmiecik at wanadoo.fr>
  • Date: Mon, 15 Sep 2008 03:42:03 -0400 (EDT)

Hello to all

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) &
> 
> 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}}
> 
> Out[14]= {0}

but not this last one :

> In[16]:=      FullSimplify[Subtract @@ {#[[1]][x], #[[2]][Tanh x/2]}, 
>                        x\[Epsilon] Reals] & /@ {{Gd , 2 ArcTan}}
> 
> Out[16]=    {-\[Pi]/2 + 
>                         2 ArcTan[\[ExponentialE]^x] - (2 ArcTan)[(Tanh x)/2]}

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

Thanks

Sigismond Kmiecik








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