Singularities and isolated singularities. Residue of a function at an isolated singularity. Residue theorem. Residue theorem with the residue at infinity. Classification of isolated singularities: removable singularities, poles of order m, essential singularities. Necessary and sufficient condition for an isolated singular point of an analytic function to be a pole of order m and formula for the corresponding residue. Zeros of order m of analytic functions. Necessary and sufficient condition for an analytic function to have a zero of order m. Identity theorem. The zeros of non-constant analytic functions are isolated and of finite order. Sufficient condition for a function of the type \(f(z)=p(z)/q(z)\) to have a pole of order m and formula for the corresponding residue. Behavior of an analytic function in the vicinity of isolated singularities. Riemann lemma. Casorati-Weierstrass theorem.