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Re: Real and Complex Roots presented in a single plot

• To: mathgroup at smc.vnet.net
• Subject: [mg91860] Re: [mg91820] Real and Complex Roots presented in a single plot
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Wed, 10 Sep 2008 05:10:32 -0400 (EDT)
• References: <200809091056.GAA27909@smc.vnet.net>

Narasimham wrote:
> z[x_] = 1.3  Sin[1.7* x] + 0.6 Sin[4* x] ;
> Plot[z[x], {x, 0, 18}]
> 
> In the above plot we can see all the real roots of z very
> approximately at {2.1,3.4, 5.5, 7.6, 9, 11, 13.1, 14.5, 16.5}, as the
> curve crosses x-axis.
> 
>          We can also recognize and see the real parts of all the
> complex roots of z where the curve is nearest to x -axis at  {1.4,
> 4.2, 6.6, 9.6, 12.3, 15.3, 17.7}. They are near to x-values where the
> local maxima/minima occur.But we cannot 'see' their complex parts, as
> they need to be computed.
> 
> After computation of all real and complex roots I would like  to
> represent all roots,  real and complex  roots alike, on an ( x - y )
> Argand diagram (in a 2D plot)  with the real part on x - axis and
> imaginary part on y - axis.   How to do this ?
> 
>   Regards,
>   Narasimham

This computes all roots with nonnegative imaginary part. You can flip 
across the real axis to get the conjugates.

ListPlot[
  Map[{Re[#], Im[#]} &,
   x /. N[{ToRules[
       Reduce[{z[x] == 0, -100 <= Re[x] <= 100, 0 <= Im[x] <= 1/2},
        x]]}]]]

I capped the imaginary part at 1/2 because FindInstance tells me I can 
do so.

In[27]:= FindInstance[{z[x] == 0, Im[x] > 1/2}, x]
Out[27]= {}

You can also get a hint of this from plotting the absolute value, 
letting the imaginary part of x go from 0 to 1.

Plot3D[Abs[z[x + I*y]], {x, 0, 18}, {y, 0, 1}]

Daniel Lichtblau
Wolfram Research

• References:
  • Real and Complex Roots presented in a single plot
    • From: Narasimham <mathma18@hotmail.com>