I was reading about Laurent series from the wikipedia and in one of the conditions of Laurent series it is mentioned that
On the boundary of the annulus, one cannot make a general statement, except to say that there is at least one point on the inner boundary and one point on the outer boundary such that f(z) cannot be holomorphically continued to those points.
as I don't know much about complex analysis other than some basic stuff
Is "holomorphically continued" used there is same as analyticity at the boundary? or it is something else and if it is then please anyone explain it in layman's terms.
"Holomorphically continued" means that if you have your Laurent series $f(z)$ in the annulus $|z-z_0| < R$, you can define a function $\tilde{f}(z)$ on some set $S$ that intersects the annulus, such that $\tilde{f}(z)$ is holomorphic.
It is a similar idea to how you can treat the (shifted) gamma function as an analytic continuation of the factorial function - the factorial is defined only on the natural numbers, but the gamma function is analytic on (most of) the complex numbers.