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Re: Forcing Trig Identities

  • To: mathgroup at smc.vnet.net
  • Subject: [mg94622] Re: [mg94615] Forcing Trig Identities
  • From: Carl Woll <carlw at wolfram.com>
  • Date: Thu, 18 Dec 2008 07:19:39 -0500 (EST)
  • References: <200812171136.GAA09099@smc.vnet.net>

Gary McClelland wrote:

>In earlier versions of Mathematica (I think 5 and before), the following code:
>
>pmn[\[Rho]_ , x_, z_] := 
> PDF[MultinormalDistribution[{0, 0}, {{1, \[Rho]}, {\[Rho], 1}}], {x, 
>   z}]
>Integrate[pmn[\[Rho], x, z], {x, -Infinity, 0}, {z, -Infinity, 0}, 
> Assumptions -> {\[Rho] ^2 < 1, \[Rho] > 0}]
>
>yielded:
>(\[Pi] + 2 ArcSin[\[Rho]])/(4 \[Pi])
>
>now (Mathematica 7), I get the mathematically equivalent but less desirable:
>(\[Pi] - ArcTan[Sqrt[-1 + 1/\[Rho]^2]])/(2 \[Pi])
>
>TrigReduce and FullSimplify fail to simplify what Mathematica 7 gives to the simpler expression Mathematica 5 yielded.  any suggestions on how to force trig identities?
>  
>
Here is one method:

In[17]:= Assuming[\[Rho]>0,(\[Pi]-ArcTan[Sqrt[-1+1/\[Rho]^2]])/(2 
\[Pi])//FunctionExpand//FullSimplify]
Out[17]= (\[Pi]+2 ArcSin[\[Rho]])/(4 \[Pi])

Carl Woll
Wolfram Research


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