Re: plotting complex functions in (x,y,t) space

• To: mathgroup at smc.vnet.net
• Subject: [mg132150] Re: plotting complex functions in (x,y,t) space
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Fri, 3 Jan 2014 04:43:07 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• Delivered-to: l-mathgroup@wolfram.com
• Delivered-to: mathgroup-outx@smc.vnet.net
• Delivered-to: mathgroup-newsendx@smc.vnet.net
• References: <20131006074734.D77A66A1D@smc.vnet.net>

```I recommend a different approach. Plot contours of the magnitude with the
coloring set by the argument.

psin[x_, xi_, y_, yi_, t_, ti_, kx_, ky_] =
(20 E^(1/2 I kx (-kx (t - ti) + 2 (x - xi)) +
1/2 I ky (-ky (t - ti) + 2 (y - yi)) -
((-kx (t - ti) + (x - xi))^2 +
(-ky (t - ti) + (y - yi))^2)/
(1600 + 2 I (t - ti))) Sqrt[2/=F0])/
(800 + I (t - ti));

f[x_, y_, t_] =
psin[x, 0, y, 0, t, 0, 2/10, 2/10];

The magnitude of f is given by

absf[x_, y_, t_] = ComplexExpand[Abs[f[x, y, t]],
TargetFunctions -> {Re, Im}]

(20*E^(-((1600*(-(t/5) + x)^2)/(2560000 + 4*t^2)) -
(1600*(-(t/5) + y)^2)/(2560000 + 4*t^2))*Sqrt[2/Pi])/
Sqrt[640000 + t^2]

The minimum magnitude is close to zero.

magMin = With[{
xmin = -125, ymin = -125, tmin = 0,
xmax = 200, ymax = 200, tmax = 200},
Minimize[{absf[x, y, t],
xmin <= x <= xmax, ymin <= y <= ymax, tmin <= t <= tmax},
{x, y, t}][[1]] // Simplify]

1/(20*E^50*Sqrt[2*Pi])

The maximum magnitude is

magMax = With[{xmin = -125, ymin = -125, tmin = 0,
xmax = 200, ymax = 200, tmax = 200},
Maximize[{absf[x, y, t],
xmin <= x <= xmax, ymin <= y <= ymax, tmin <= t <= tmax},
{x, y, t}][[1]] // Simplify]

1/(20*Sqrt[2*Pi])

The argument of f is

argf[x_, y_, t_] = ComplexExpand[Arg[f[x, y, t]],
TargetFunctions -> {Re, Im}] // Simplify

Rewriting argf

argf2[x_, y_, t_] = Module[
{z, p, q = (640000 + t^2)},
z = (256000*(x + y) +
t*(-51200 + x^2 + y^2))/(2*q);
p = -(16*(2*t^2 - 10*t*
(x + y) + 25*(x^2 + y^2)))/q;
ArcTan[(E^p*(800*Cos[z] + t*Sin[z]))/q,
(E^p*((-t)*Cos[z] + 800*Sin[z]))/ q]];

Verifying that the expressions are the same

argf[x, y, t] == argf2[x, y, t]

True

The mnimum is determined numerically to be -Pi

argMin = With[{
xmin = -125, ymin = -125, tmin = 0,
xmax = 200, ymax = 200, tmax = 200},
NMinimize[{argf[x, y, t],
xmin <= x <= xmax, ymin <= y <= ymax, tmin <= t <= tmax},
{x, y, t}][[1]]]

-3.14159

And the maximum is Pi

argMax = With[{
xmin = -125, ymin = -125, tmin = 0,
xmax = 200, ymax = 200, tmax = 200},
NMaximize[{argf[x, y, t],
xmin <= x <= xmax, ymin <= y <= ymax, tmin <= t <= tmax},
{x, y, t}][[1]]]

3.14159

Use Manipulate to vary the magnitude of the contour. This is quite slow due
to the complexity of the functions involved.

With [{step = (magMax - magMin)/100.},
Manipulate[
ControlActive[
(c - magMin)/(magMax - magMin),
Module[{
xmin = -125., ymin = -125., tmin = 0.,
xmax = 200., ymax = 200., tmax = 200.},
ContourPlot3D[absf[x, y, t] == c,
{x, xmin, xmax}, {y, ymin, ymax}, {t, tmin, tmax},
ColorFunction -> Function[{x, y, t, p},
Hue[(argMax - argf2[x, y, t])/(argMax - argMin)]],
ColorFunctionScaling -> False]]],
{{c, magMin + step, "Abs[f[x,y,t]]"},
magMin + step, magMax - step, step,
Appearance -> "Labeled"}]]

Bob Hanlon

On Sat, Dec 21, 2013 at 2:29 PM, Michael B. Heaney <mheaney at alum.mit.edu>wr=
ote:

> Hi Bob,
>
> Thanks again for your help. I have modified your code a little, see below=

```

• Prev by Date: Re: Step-By-Step Solutions No Longer working?
• Next by Date: Re: For 2014?
• Previous by thread: Re: Step-By-Step Solutions No Longer working?
• Next by thread: Re: For 2014?