2
$\begingroup$

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?

$\endgroup$
3
  • 1
    $\begingroup$ Please explain what's the meaning of asserting that two functions have the same singularities. $\endgroup$ Sep 26 '18 at 10:15
  • 3
    $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.