Re: ideas please!
- To: mathgroup at smc.vnet.net
- Subject: [mg78896] Re: [mg78850] ideas please!
- From: DrMajorBob <drmajorbob at bigfoot.com>
- Date: Thu, 12 Jul 2007 05:18:23 -0400 (EDT)
- References: <8944432.1184150461432.JavaMail.root@m35>
- Reply-to: drmajorbob at bigfoot.com
There's a branch cut of Log where its argument crosses the x-axis:
imLogArg =
Im[logArg =
1 - (\[ImaginaryI] + z)/((-(1/3) + \[ImaginaryI]) +
1/6 (1 + \[ImaginaryI] Sqrt[3]) (1/2 (29 - 3 Sqrt[93]))^(
1/3) + (1 - \[ImaginaryI] Sqrt[3])/(
3 2^(2/3) (29 - 3 Sqrt[93])^(1/3)))] // ComplexExpand;
soln = Solve[imLogArg == 0, z];
z /. First@soln // Simplify
N@%
(2 2^(1/3) + (58 - 6 Sqrt[93])^(2/3) -
4 (29 - 3 Sqrt[93])^(1/3))/(-2 2^(1/3) Sqrt[3] +
Sqrt[3] (58 - 6 Sqrt[93])^(2/3) + 12 (29 - 3 Sqrt[93])^(1/3))
1.12214
and this is Log's argument at that point:
logArg // ComplexExpand;
% /. soln // Simplify
% // N
Log[%]
{(2^(1/3) Sqrt[
3] (-2 + 2^(1/3) (29 - 3 Sqrt[93])^(2/3)))/(-2 2^(1/3) Sqrt[3] +
Sqrt[3] (58 - 6 Sqrt[93])^(2/3) + 12 (29 - 3 Sqrt[93])^(1/3))}
{-3.82048}
{1.34038+ 3.14159 \[ImaginaryI]}
Here are the approximate values left and right of the jump:
Log[logArg] /. z -> jump - .00001
Log[logArg] /. z -> jump + .00001
1.34037- 3.14159 \[ImaginaryI]
1.34038+ 3.14159 \[ImaginaryI]
The imaginary parts are +/- Pi I.
(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,
1.1, 1.2}, PlotRange -> All] &) /@ {Re, Im}
Bobby
On Wed, 11 Jul 2007 05:09:16 -0500, dimitris <dimmechan at yahoo.com> 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
>
>
>
-- =
DrMajorBob at bigfoot.com