Re: ideas please!
- To: mathgroup at smc.vnet.net
- Subject: [mg78894] Re: [mg78850] ideas please!
- From: Daniel Lichtblau <danl at wolfram.com>
- Date: Thu, 12 Jul 2007 05:17:19 -0400 (EDT)
- References: <200707111009.GAA05013@smc.vnet.net>
dimitris wrote:
> Hello.
>
> In[201]:=
> (Plot[#1[Log[1 - (I + z)/((-(1/3) + I) + (1/6)*(1 +
> I*Sqrt[3])*((1/2)*(29 - 3*Sqrt[93]))^(1/3) +
> (1 - I*Sqrt[3])/(3*2^(2/3)*(29 - 3*Sqrt[93])^(1/3)))]], {z,
> 0, 5}] & ) /@ {Re, Im};
>
> As it is clear from the graph the Imaginary part has a jump
> discontinuity somewhere near z=1.
> Can somebody point me out a way to detect exactly its position?
> No matter what I have tried I had no success.
>
> Dimitris
>
Solve for where argument of log intersects log branch cut. Reduce will
handle this task.
logexpr = Log[1 - (I + z)/((-(1/3) + I) +
(1/6)*(1 + I*Sqrt[3])*((1/2)*(29 - 3*Sqrt[93]))^(1/3) +
(1 - I*Sqrt[3])/(3*2^(2/3)*(29 - 3*Sqrt[93])^(1/3)))];
arg = First[logexpr];
repart = ComplexExpand[Re[arg]];
impart = ComplexExpand[Im[arg]];
InputForm[crossing = Reduce[{repart<0, impart==0, 0<=z<=5}, z]]
Out[6]//InputForm=
z == (-2*2^(1/3) - Root[256 - 13424*#1^3 + #1^6 & , 1, 0] +
4*Root[4 - 58*#1^3 + #1^6 & , 1, 0])/(2*2^(1/3)*Sqrt[3] -
Sqrt[3]*Root[256 - 13424*#1^3 + #1^6 & , 1, 0] -
12*Root[4 - 58*#1^3 + #1^6 & , 1, 0])
N[crossing]
Out[7]= z == 1.12214
Daniel Lichtblau
Wolfram Research
- References:
- ideas please!
- From: dimitris <dimmechan@yahoo.com>
- ideas please!