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