Re: Simplifying expression involving Log and I
- To: mathgroup at smc.vnet.net
- Subject: [mg38043] Re: Simplifying expression involving Log and I
- From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
- Date: Tue, 26 Nov 2002 00:51:00 -0500 (EST)
- Organization: Universitaet Leipzig
- References: <arsit4$ef4$1@smc.vnet.net>
- Reply-to: kuska at informatik.uni-leipzig.de
- Sender: owner-wri-mathgroup at wolfram.com
Hi,
try
In[]:=expr = d*Pi + I*d*(Log[M] - 2*Log[(-I)*Sqrt[(M*r)/d] + Sqrt[M -
(M*r)/d]]);
In[]:=expr1=expr /.
a_.*M + b_*M :> M*(a + b) /.
Sqrt[M*a_] :> Sqrt[M]*Sqrt[a] /.
a_.Log[b_] + c_.*Log[d_] :> Log[b^a*d^c] /.
Sqrt[M]*a_ + Sqrt[M]*b_ :> Sqrt[M]*(a + b) /.
Log[a_^b_] :> b*Log[a] /.
Log[-I*a_ + b_] :> Log[-I*(a + I*b)] /.
Log[a_*b_Plus] :> Log[a] + Log[b]//Expand
Out[]=(-2*I)*d*Log[Sqrt[r/d] + I*Sqrt[1 - r/d]]
In[]:=FullSimplify[expr1, Element[{r, d}, Reals] && r < d]
Out[]=2*d*ArcSin[Sqrt[1 - r/d]]
ups...
Regards
Jens
Dave Snead wrote:
>
> Hi,
>
> I'm trying to simplify (M, r, d are positive reals with r<d)
> d*Pi + I*d*(Log[M] - 2*Log[(-I)*Sqrt[(M*r)/d] + Sqrt[M - (M*r)/d]])
>
> to
>
> 2*d*ArcCos[Sqrt[r/d]]
>
> I've tried applying
>
> FullSimplify[#,M>0 && 0<r<d]&
>
> and
>
> ExpToTrig
>
> but Mathematica won't get rid of the Log and the I and give me the ArcCos.
>
> Does anyone know which Mathematica functions to apply to my expression to
> give me the simplification I want?
>
> Thanks in advance.