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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.2012-12-28
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    @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.2012-12-28
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    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 get2012-12-28
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    sounds good. Thanks for the suggestion.2012-12-28

4 Answers 4

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

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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.2012-12-28
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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!2013-05-10