Series and residues

Concepts

• Complex Taylor series 10 exercisesThe Taylor series of a function f holomorphic on a disc |z-z_0|<R is the power series in z-z_0, centred at z_0, whose coefficient…
• Laurent series 10 exercisesThe Laurent series of a function f holomorphic on an annulus r<|z-z_0|<R is its expansion in integer powers, positive and negativ…
• Isolated singularity 10 exercisesAn isolated singularity of a function f is a point z_0 where f is not holomorphic but is holomorphic on a punctured disc 0<|z-z_0…
• Zero and pole 10 exercisesA zero of order m of a holomorphic function f is a point z_0 where f vanishes with multiplicity m, with holomorphic quotient g; a…
• Residue 10 exercisesThe residue of a function f at an isolated singularity z_0 is the coefficient a_-1 of (z-z_0)^-1 in its Laurent series on a punct…
• Residue theorem 10 exercisesIf f is holomorphic on a simple closed contour γ and inside it except at isolated singularities z_k, its integral over γ is 2π i…
• Real integral by residues 10 exercisesA method to compute ∫_-∞^∞ f(x) dx with f rational: close the real axis with a semicircle in the upper half-plane and apply the r…