0
$\begingroup$

I am calculating some residue calculus stuff, where I need to know if the prescribed poles are inside the curve given above, namely $2\sin(2\theta)$ for $0\leq \theta<2\pi$. I actually need to know if these are $( e^{(\pi i)/4}, e^{(3\pi i)/4},e^{(5\pi i)/4},e^{(7\pi i)/4})$inside the curve or not?

I am actually very bad at graphing, any help will be much appreciated.

  • 0
    sounds good. Thanks for the suggestion.2012-12-28

4 Answers 4

1

You Can use Matlab Program

 t=0:pi/100:2*pi;  r=2*sin(2*t);  p1=polar(t,r)  set(p1,'Linewidth',2);  hold on  z1=exp(pi*i/4);  z2=exp(3*pi*i/4);  z3=exp(5*pi*i/4);  z4=exp(7*pi*i/4);  z=[z1 z2 z3 z4];  p2=polar( angle(z),abs(z),'rx');  set(p2,'LineWidth',12)  hold off  legend('r=2sin(2\theta)','points',-1) 

enter image description here

also you can use mathematica

   Show[PolarPlot[2 Sin[2 t], {t, 0, 2 Pi}],      ListPolarPlot[{{Pi/4, 1}, {3 Pi/4, 1}, {5 Pi/4, 1}, {7 Pi/4, 1}}]] 

enter image description here

3

Hint: $e^{pi x}$ has $\theta = x$ and $r = 1$. What is $\sin(2 \theta)$ when $\theta = \pi/4$, say?

2

I suggest using FooPlot (free and online). Here is the graph of your polar function. Can you decide now which poles are in the region bounded by the contour? If you want to learn how to graph polar functions yourself I would suggest reading this

  • 0
    I think all of the poles inside the graph. This helps a lot. Thanks for the site and plot.2012-12-28
2

You can use WolframAlpha for the plot.

Additionally, you can use almost any Computer Algebra System (CAS) and you can find a CAS List here.

Regards

  • 0
    Short but in$f$ormative!2013-05-10