When estimating asymptotics of a series from its generating function, we look for singularities (this makes sense to me) and then try to remove them (this also makes sense) by subtracting a function which has the same singularities (i.e. they have a singularity at the same set of points). This I do not understand. If I have functions $f$ and $g$ with the same singularities, why does $f - g$ have no singularities? Why is this the case? Or am I missing something and it is wrong?

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    $\begingroup$ Please explain what's the meaning of asserting that two functions have the same singularities. $\endgroup$ Sep 26 '18 at 10:15
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    $\begingroup$ Are you familiar with Laurent series? If you have two functions with the same 'negative part' you get a power series when you subtract. $\endgroup$ Sep 26 '18 at 10:18
  • $\begingroup$ Does this mean, that if they have the same singularities, they must have the same principal part and, therefore, I get a function with no principal part when I subtract them? If so, why is this the case? $\endgroup$ Sep 26 '18 at 10:23

Take $f(z)=\dfrac{1}{z}$ and $g(z)=\dfrac{1}{z}-1$. They both have a singularity at $0$. Now what is $f(z)-g(z)$? You'll see that you don't have a singularity anymore.

Now take $f(z)=\dfrac{1}{z}$ and $g(z)=-\dfrac{1}{z}-1$ or $g(z)=\dfrac{1}{z^2}$. They both have a singularity at $0$. What is $f(z)-g(z)$? You'll see that your singularity is not removed.

If you have two holomorphic functions $f$ and $g$ on some domain, to investigate the singularities of $f-g$, use their Laurent series.


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