       Re: Function Equivalence of ArcTan and Log

• To: mathgroup at smc.vnet.net
• Subject: [mg114331] Re: Function Equivalence of ArcTan and Log
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Wed, 1 Dec 2010 02:12:06 -0500 (EST)

```The equation s1==s2 need not be true!  For example:

s1 /. {x->0, y->2} //InputForm
-Pi - I*Log

s2 /. {x->0, y->2} //FullSimplify//InputForm
(-I)*Log

On 11/22/2010 7:38 AM, Dr. Robert Kragler wrote:
>    Hi,
>
> I have a problem to show with Mathematica (using already TrigToExp and
> ComplexExpand) that
>
>          In:= s1=(ArcTan[x-I y]-ArcTan[x+I y])//TrigToExp
>          Out:= 1/2 I Log[1-I (x-I y)]-1/2 I Log[1+I (x-I y)]-1/2 I Log[1-I
> (x+I y)]+1/2 I Log[1+I (x+I y)]
> and
>
>          In:= s2=I/2 Log[(x^2+(1-y)^2)/(x^2+(1+y)^2)]//ComplexExpand
> Out:= I (1/2 Log[x^2+(1-y)^2]-1/2 Log[x^2+(1+y)^2])
>
> are identical, i.e. s1===s2 should give "True". But FullSimplify applied to lhs
> or rhs does not help.
> Of course, the identity wanted can be derived by hand making use of the inverse
> function Tan[s1]==I z and Sin[s1], Cos[s1] etc.
> which gives rise to the following replacement rule :
>
> ArcTanToLog = {(ArcTan[x_-I y_]-ArcTan[x_+I y_])->  I/2
> Log[(x^2+(1-y)^2)/(x^2+(1+y)^2)]}
>
> It is, however, non-trivial to prove this identity purely with term rewriting by
> means of Mathematica.
>
> Any suggestion is appreciated. Thanks in advance,
>
> Robert Kragler
>
>

--
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

```

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