Complex differentiability; differentiability properties of elementary functions and harmonic functions; the reflection principle; integration along contours; the Cauchy-Goursat theorem; simple and multiple connectedness of domains; the Cauchy integral formula; the maximum modulus principle; Taylor and Laurent series; analytic continuation; absolute and uniform series convergence; classification of isolated singularities; branch cuts; zeroes and poles; residues; the Cauchy residue theorem and Jordan's lemma; principal values; computation of definite and improper integrals involving polynomials and trigonometric functions; contour deformation around branch points and branch cuts; Rouché's theorem.