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verification

• To: mathgroup at smc.vnet.net
• Subject: [mg74701] verification
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Sun, 1 Apr 2007 04:15:39 -0400 (EDT)

Hello.


foo = {ArcTan[8/(1 - Sqrt[-15 - 4*I])] + ArcTan[8/(1 + Sqrt[-15 -
4*I])] + ArcTan[8/(1 - Sqrt[-15 + 4*I])] +
     ArcTan[8/(1 + Sqrt[-15 + 4*I])], ArcTan[3] + ArcTan[5] +
ArcTan[41/3] + ArcTan[21], 2*Pi - ArcTan[1/4] - ArcTan[5/12]};

The elements of foo list are equal

Chop[N[foo, 30]]
{5.64341552435296080601310475496,5.64341552435296080601310475496,5.\
64341552435296080601310475496}

Block[{Message}, FullSimplify[foo[[2]] == foo[[3]]]]
Block[{Message}, FullSimplify[foo[[1]] == foo[[3]]]]
Block[{Message}, FullSimplify[foo[[1]] == foo[[2]]]]

True
ArcCot[4] + ArcTan[5/12] + ArcTan[8/(1 - Sqrt[-15 - 4*I])] + ArcTan[8/
(1 + Sqrt[-15 - 4*I])] + ArcTan[8/(1 - Sqrt[-15 + 4*I])] +
   ArcTan[8/(1 + Sqrt[-15 + 4*I])] == 2*Pi
ArcTan[8/(1 - Sqrt[-15 - 4*I])] + ArcTan[8/(1 + Sqrt[-15 - 4*I])] +
ArcTan[8/(1 - Sqrt[-15 + 4*I])] +
   ArcTan[8/(1 + Sqrt[-15 + 4*I])] == ArcTan[3] + ArcTan[5] +
ArcTan[41/3] + ArcTan[21]

In one of my attempts to show that foo[[1]]=foo[[3]] and
foo[[1]]=foo[[2]] I try

Block[{Message}, (FullSimplify[#1, ComplexityFunction -> LeafCount] & )
[foo[[1]] == foo[[3]]]]
Block[{Message}, (FullSimplify[#1, ComplexityFunction -> LeafCount] & )
[foo[[1]] == foo[[2]]]]

2*Pi + ArcTan[8/(-1 + Sqrt[-15 - 4*I])] + ArcTan[8/(-1 + Sqrt[-15 +
4*I])] ==
  ArcCot[4] + ArcTan[5/12] + ArcTan[8/(1 + Sqrt[-15 - 4*I])] +
ArcTan[8/(1 + Sqrt[-15 + 4*I])]
ArcTan[8/(1 + Sqrt[-15 - 4*I])] + ArcTan[8/(1 + Sqrt[-15 + 4*I])] ==
ArcTan[3] + ArcTan[5] + ArcTan[41/3] + ArcTan[21] + ArcTan[8/(-1 +
Sqrt[-15 - 4*I])] + ArcTan[8/(-1 + Sqrt[-15 + 4*I])]

but I failed.

Introducing the following ComplexityFunction

lst = Alternatives @@ Replace[ToExpression[Names["Arc*"]], x_ -> _x,
-1]
_ArcCos | _ArcCosh | _ArcCot | _ArcCoth | _ArcCsc | _ArcCsch | _ArcSec
| _ArcSech | _ArcSin | _ArcSinh | _ArcTan | _ArcTanh

I got

TimeConstrained[Block[{Message}, (FullSimplify[#1, ComplexityFunction -
> (Count[{#1}, lst, Infinity] & )] & )[
    foo[[1]] == foo[[3]]]], 300]
$Aborted

TimeConstrained[Block[{Message}, (FullSimplify[#1, ComplexityFunction -
> (Count[{#1}, lst, Infinity] & )] & )[
    foo[[1]] == foo[[2]]]], 300]
0 == 4*Pi + 2*Log[(825/2873 - (2752*I)/2873)^(-(I/2))] + I*Log[-(((-1
- 8*I) + Sqrt[-15 - 4*I])/(1 - Sqrt[-15 - 4*I]))] -
   I*Log[-(((-1 + 8*I) + Sqrt[-15 - 4*I])/(1 - Sqrt[-15 - 4*I]))] -
I*Log[((1 - 8*I) + Sqrt[-15 - 4*I])/(1 + Sqrt[-15 - 4*I])] +
   I*Log[((1 + 8*I) + Sqrt[-15 - 4*I])/(1 + Sqrt[-15 - 4*I])] + I*Log[-
(((-1 - 8*I) + Sqrt[-15 + 4*I])/(1 - Sqrt[-15 + 4*I]))] -
   I*Log[-(((-1 + 8*I) + Sqrt[-15 + 4*I])/(1 - Sqrt[-15 + 4*I]))] -
I*Log[((1 - 8*I) + Sqrt[-15 + 4*I])/(1 + Sqrt[-15 + 4*I])] + I*Log[((1
+ 8*I) + Sqrt[-15 + 4*I])/(1 + Sqrt[-15 + 4*I])]


Having failed also in similar attempts I would really appreciate any
ideas!

Dimitris

• Follow-Ups:
  • Re: verification
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>