# Why does subtracting function with the same singularities make it analytic

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?

• Please explain what's the meaning of asserting that two functions have the same singularities. Sep 26 '18 at 10:15
• Are you familiar with Laurent series? If you have two functions with the same 'negative part' you get a power series when you subtract. Sep 26 '18 at 10:18
• 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? 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.