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Complex Roots

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  Complex Roots Joel <joel.steenis@gmail.com> - 2011-07-01 08:25 -0700
    Re: Complex Roots Joe Riel <joer@san.rr.com> - 2011-07-01 10:39 -0700
      Re: Complex Roots A N Niel <anniel@nym.alias.net.invalid> - 2011-07-02 06:46 -0600

#197 — Complex Roots

I am trying to find all the complex roots of the equation T given
below:

w := 60*(2*3.14159);
R12 := .5;
L12 := 3/w;
Z12 := R12+s*L12+I*w*L12;
kp := 0.5e-3;
kv := 0.5e-3;
Q1 := 384;
Q2 := 375;

apw1 := s+kp*(-Q1+w*L12/abs(Z12)^2);
apw2 := s+kp*(-Q2+w*L12/abs(Z12)^2);
aqv1 := 1+kv*(Q1+w*L12/abs(Z12)^2);
aqv2 := 1+kv*(Q2+w*L12/abs(Z12)^2);
T := apw1*apw2*aqv1*aqv2;


I have used the solve, fsolve (with complex argument), and RootOf
function but all that is returned is one real root or nothing at all.

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#198

Joel <joel.steenis@gmail.com> writes:

> I am trying to find all the complex roots of the equation T given
> below:
>
> w := 60*(2*3.14159);
> R12 := .5;
> L12 := 3/w;
> Z12 := R12+s*L12+I*w*L12;
> kp := 0.5e-3;
> kv := 0.5e-3;
> Q1 := 384;
> Q2 := 375;
>
> apw1 := s+kp*(-Q1+w*L12/abs(Z12)^2);
> apw2 := s+kp*(-Q2+w*L12/abs(Z12)^2);
> aqv1 := 1+kv*(Q1+w*L12/abs(Z12)^2);
> aqv2 := 1+kv*(Q2+w*L12/abs(Z12)^2);
> T := apw1*apw2*aqv1*aqv2;
>
>
> I have used the solve, fsolve (with complex argument), and RootOf
> function but all that is returned is one real root or nothing at all.

It only has two roots, both real.  That is apparent from the 
form of the expressions.  One way to produce both of them is

map(fsolve, {op(T)});

-- 
Joe Riel

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#199

In article <87sjqp294m.fsf@san.rr.com>, Joe Riel <joer@san.rr.com>
wrote:

> Joel <joel.steenis@gmail.com> writes:
> 
> > I am trying to find all the complex roots of the equation T given
> > below:
> >
> > w := 60*(2*3.14159);
> > R12 := .5;
> > L12 := 3/w;
> > Z12 := R12+s*L12+I*w*L12;
> > kp := 0.5e-3;
> > kv := 0.5e-3;
> > Q1 := 384;
> > Q2 := 375;
> >
> > apw1 := s+kp*(-Q1+w*L12/abs(Z12)^2);
> > apw2 := s+kp*(-Q2+w*L12/abs(Z12)^2);
> > aqv1 := 1+kv*(Q1+w*L12/abs(Z12)^2);
> > aqv2 := 1+kv*(Q2+w*L12/abs(Z12)^2);
> > T := apw1*apw2*aqv1*aqv2;
> >
> >
> > I have used the solve, fsolve (with complex argument), and RootOf
> > function but all that is returned is one real root or nothing at all.
> 
> It only has two roots, both real.  That is apparent from the 
> form of the expressions.  One way to produce both of them is
> 
> map(fsolve, {op(T)});

More details.  The complex roots of the product T are the complex roots
of the factors apw1, apw2, aqv1, aqv2.  Factors aqv1 and aqv2 are
always positive, and therefore have no complex roots.  Factors apw1 and
apw2 have imaginary part Im(s), so all complex roots are actually real.
Finally, each of these factors has one real root,
0.1918378646 and 0.1873378640 as shown.

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