"A point $z$ is said to be a singularity of the function $F(z)$ if in the complex plane there exists no circle with center at $z$ within which $F(z)$ is analytic."
Can someone describe this a little better to me? Just having trouble grasping the definition given above.
This definition says that a point $z_0$ is a singularity if the function $F$ isn't holomorphic in the point $z_0$. Note that a complex function is holomorphic in $z_0$ if and only if it is analytic in this point. For example you can take the function $f(z)=\dfrac{1}{z}$. This function has a syngualrity in $z_0=0$.