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