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Re: Simplification and Arg[]


Andrew Moylan wrote:
> Should Mathematica be able to simplify the following expression? (It is 
> easily seen to be zero under the given condition, x > 0.)
> 
> FullSimplify[
> 	Arg[1 + I * x] + Arg[1 - I * x],
> 	{x > 0}
> ]
> 
> In particular, I would have expected the following to yield ArcTan[b / 
> a], from which the above expression is easily reduced to zero:
> 
> FullSimplify[
> 	Arg[a + I b],
> 	{a > 0, b > 0}
> ]
> 
> Any ideas?
> 
> Cheers,
> 
> Andrew
> 
> P.S. Apologies if I have sent this twice; my original message seems not 
> to have worked.
> 
Andrew Moylan wrote:
 > Should Mathematica be able to simplify the following expression? (It is
 > easily seen to be zero under the given condition, x > 0.)
 >
 > FullSimplify[
 >       Arg[1 + I * x] + Arg[1 - I * x],
 >       {x > 0}
 > ]

Hi Andrew,

What you need is *ComplexExpand* [1]:

In[1]:=
ComplexExpand[Arg[1 - I*x] + Arg[1 + I*x],
   TargetFunctions -> {Re, Im}]

Out[1]=
0

 > In particular, I would have expected the following to yield ArcTan[b /
 > a], from which the above expression is easily reduced to zero:
 >
 > FullSimplify[
 >       Arg[a + I b],
 >       {a > 0, b > 0}
 > ]

In[2]:=
ComplexExpand[Arg[a + I*b], TargetFunctions ->
    {Re, Im}]

Out[2]=
ArcTan[a, b]

In[3]:=
FullSimplify[ComplexExpand[Arg[a + I*b], {a, b},
    TargetFunctions -> {Re, Im}], {a > 0, b > 0}]

Out[3]=
        b
ArcTan[-]
        a

For the difference between Out[2] (without information on the quadrant)
and Out[3] (with information on the quadrant) see [2].

Best regards,
Jean-Marc

[1] http://documents.wolfram.com/mathematica/functions/ComplexExpand

[2] http://documents.wolfram.com/mathematica/functions/ArcTan


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