A function is analytic in a circle

Question

I know that a function is analytic if it satisfies the Cauchy Riemann rules but how can my professor can tell without doing any kind of calculation that a function is analytic in a circle? As an example: $f(z): \frac{\sin(hz)}{z^3}$ is analytic in $C:||z-2||=1$ therefore the integral of f in C is zero.