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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.

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please suggest some sites where I can graph these sort of functions. –  Deepak Dec 28 '12 at 7:44
3  
I would suggest increasing your accept rate. 54% may turn people off. –  Nameless Dec 28 '12 at 7:48
    
@Nameless, I try to accept all the answers, I accept those which completely make sense to me otherwise I would take time to understand and re-ask. I just don't accept answers sometime thinking that somebody will be there with better and easy approach. I am not trying to be rude, but just don't want to show myself oversmart by accepting the fact which I can not follow. –  Deepak Dec 28 '12 at 16:30
    
I understand your point. Of course you should accept the posts that you can understand and follow, at your own time. 79% is a good percentage. Try to keep it around 80% and people will know that you care about the answers you get –  Nameless Dec 28 '12 at 17:49
    
sounds good. Thanks for the suggestion. –  Deepak Dec 28 '12 at 17:50

4 Answers 4

up vote 1 down vote accepted

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

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I think all of the poles inside the graph. This helps a lot. Thanks for the site and plot. –  Deepak Dec 28 '12 at 16:33

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

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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

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Short but informative! –  amWhy May 10 '13 at 1:18

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

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