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I am trying to understand the attached:

enter image description here

I know that if two functions have zeros and poles at the same point and of the same order then they differ only by a multiplicative constant, so that is fine, as both have a double zero at $$z=w_j/2$$ and a double pole at $$z=0$$.

But I don't understand at all the idea before determining what the constant $$C$$ should be? I thought that perhaps we had set the residues at the double pole $$z=0$$ equal, but this is given by:

$$\frac{1}{2}lim_{z \to 0} \frac{d}{dz}(z^2f(z)) $$,

whereas it looks like we've compared

$$lim_{z \to 0} z^{2} f(z) $$,

so unless we have some reason to take the derivative outside the limit or something, I don't understand what we've done, and even whether my thoughts are on the right track and the residues are being compared?

Many thanks in advance.

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Just bring the fraction of the RHS to the LHS such that only $C$ remains on the RHS. Multiply both the nominator and the denominator of the LHS by $z^2$ and take the limit $z\rightarrow 0$. Then you get $$C=\frac{1}{-1}=-1.$$

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  • $\begingroup$ @DonAntonio Thanks :) $\endgroup$ – Severin Schraven Apr 3 '17 at 20:11
  • $\begingroup$ Thanks. I guess what I'm trying to get at is what is the significance of taking the limit at this point $z=0$? So we are not comparing the residues just continuity at $z=0$, could we have chosen any point? Thanks $\endgroup$ – yourlazyphysicist Apr 4 '17 at 10:59
  • $\begingroup$ Any point you know the limit of both the functions would do the job. $\endgroup$ – Severin Schraven Apr 4 '17 at 20:17
  • $\begingroup$ ahh okay thanks, and $z=0$ is the easiest to work with here? $\endgroup$ – yourlazyphysicist Apr 5 '17 at 10:29
  • $\begingroup$ It has to be multiplied by at least $z^2$ though because that is the order of the pole and any power smaller the limit would be undefined? $\endgroup$ – yourlazyphysicist Apr 7 '17 at 16:47

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