Re: Simplification and Arg[]

• To: mathgroup at smc.vnet.net
• Subject: [mg66727] Re: Simplification and Arg[]
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sat, 27 May 2006 21:03:56 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <e4r7qm\$mmh\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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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