Need help to plot Talbot's Contour (Newbie question)

Question

I would like to plot these following parametrizations:

$$s(\theta)=\sigma+\lambda s_{\nu}(\theta),\quad s_{\nu}(\theta)=\theta\cot(\theta) + i\nu\theta,\quad \theta\in(-\pi,\pi)$$

For more details, please see: The accurate numerical inversion of Laplace Transform by A. Talbot. The parameters $\sigma$, $\lambda$, and $\nu$ are adjustable. The Talbot's contour should looks like these two: Or

My attempt:

σ := 1
λ := 1/2
ν := 1
s1 := θ*Cot (θ) + (I ν θ)
eqn := σ + (λ s1), Assumptions -> -Pi <= θ <= Pi
ParametricPlot[{Re[eqn], Im[eqn]}, {f, -5, 2}, AspectRatio -> 1]

Sorry, I can't give further attempt since I'm not really familiar with Mathematica. Hope you can help me. Thanks in advance.

score 4 · Accepted Answer · answered Feb 23 '21 at 15:18

4

Start from here:

σ = 1;
λ = 1/2;
ν = 1;
s1 = θ Cot[θ] + I ν θ;
eqn = σ + (λ s1);
ParametricPlot[{Re[eqn], Im[eqn]}, {θ, -3.5, 3.5}, AspectRatio -> 1]

answered Feb 23 '21 at 15:18

MarcoB

67,153
18
91
189

Wonderful. Thank you for your help!! – user516076 Feb 23 '21 at 15:20
In the current version, I would use ReIm[eqn] within ParametricPlot[]. – J. M.'s missing motivation Feb 23 '21 at 21:35

score 2 · Answer 2 · answered Feb 23 '21 at 17:05

2

Some bells and whistles.

Clear["Global`*"]

Since the "parameters ... are adjustable" use Manipulate.

Manipulate[
 Module[{s1, θ, eqn},
  s1 = θ*Cot[θ] + I*ν*θ;
  eqn = σ + (λ*s1);
  ParametricPlot[
   {Re[eqn], Im[eqn]},
   {θ, -Pi, Pi},
   PlotRange -> {{-5, 3.5}, {-4.25, 4.25}},
   AspectRatio -> 1]],
 {{σ, 1}, 0, 2, 0.01, Appearance -> "Labeled"},
 {{λ, 0.5}, 0, 2, 0.01, Appearance -> "Labeled"},
 {{ν, 1}, 0, 2, 0.01, Appearance -> "Labeled"}]

answered Feb 23 '21 at 17:05

Bob Hanlon

157,611
7
77
198

Thanks. Another wonderful solution. By the way what does Clear[Global'*"] do? – user516076 Feb 23 '21 at 21:20
1

It removes any prior definitions that might potentially interfere with the code, i.e., makes sure you are starting with a clean slate. – Bob Hanlon Feb 23 '21 at 21:23

Need help to plot Talbot's Contour (Newbie question)

2 Answers2

Linked