complex analysis explanation

Question

Can someone please explain why :

"Residue at a finite point is zero if the function is analytic at that point".

Some explanation going by the definition or Laurent's expansion will be helpful.

Answer 1

Here is the actual definition of a residue: Given a function $f$ which is holomorphic in a punctured neighborhood $\dot U(a)$, the residue of $f$ at $a$ is defined by $${\rm res}(f|a):={1\over 2\pi i}\int_{\gamma_r} f(z)\ dz\ ,$$ where $\gamma_r:=\partial D_r(a)\subset U(a)$ is the boundary of a small disk $D_r(a)$ centered at $a$.

When $f$ is analytic in all of $D_R(a)$ for some $R>0$ then Cauchy's theorem guarantees that ${\rm res}(f|a)=0$.