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FullSimplify on Hyperbolic Functions

• To: mathgroup at smc.vnet.net
• Subject: [mg53383] FullSimplify on Hyperbolic Functions
• From: carlos at colorado.edu
• Date: Sat, 8 Jan 2005 23:02:47 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

Obviously ArcCosh[1+b^2/2]=2 ArcSinh[b/2] if b>=0. Here are
4 variations on trying to show it by FullSimplify:

ClearAll[b];  r1=ArcCosh[1+b^2/2]; r2=2*ArcSinh[b/2];

Print[FullSimplify[r1-r2,b>=0]//InputForm];
ArcCosh[1 + b^2/2] - 2*ArcSinh[b/2]

Print[FullSimplify[TrigToExp[r1-r2],b>=0]//InputForm];
0

Print[FullSimplify[r1^2-r2^2,b>=0]//InputForm];
ArcCosh[1 + b^2/2]^2 - 4*ArcSinh[b/2]^2

Print[FullSimplify[TrigToExp[r1^2-r2^2],b>=0]//InputForm];
-4*Log[(b + Sqrt[4 + b^2])/2]^2 + Log[(2 + b*(b + Sqrt[4 + b^2]))/2]^2
Why only the second one works?

• Follow-Ups:
  • Re: FullSimplify on Hyperbolic Functions
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>