expansion, a generalization of the Taylor expansion involving both positive and negative powers of Z-z0 = (z-30) +>an(z-Zo The numbers an are the coefficients of the Laurent expansion of f(z) at point z= zo The first series on the right is the principal part of the Laurent expansion, and the second series is the regular part. The regular part is an ordinary power series, hence it converges in some disk 1z-zol R where R>0. Putting S=1/(2-zo), the principal part becomes >"; this power series converges for IsI p where p >0, hence the principal part converges for Iz- zol >1/p=r. When r<R, the Laurent expansion converges in th annulus r<Iz-zol R; when r>R, it diverges everywhere in the complex plane. The function f(z) has an isolated singularity at point zo if f(z) is not analyt but is analytic in the "punctured disk"0< Iz-zol R for some R>0. Isolated angularities are classified by reference to the Laurent expansion. Three types can arise 1. Removable singularity. The point zo is a removable singularity of f(z)if the principal art of the Laurent expansion of f(z) about zo is identically zero (i.e if an =0 forn=-1,-2,-3,) 2. Pole of order k. The point zo is a pole of order k if the principal part of the laurent expansion about zo contains only finitely many terms that form a polynomial of egree k in(z-z0)-. A pole of order 1 is called a simple pole 3. Essential singularity. The point zo is an essential singularity of f(z) if the principal part of the Laurent expansion of f(z)about zo contains infinitely many terms (i.e if a-n+0 for infinitely many n) The coefficient a-I in the Laurent expansion of f(z) about an isolated singular point is the residue of f(z)at zo. It can be shown that a-1 f(z)dz where r is any simple closed curve oriented counterclockwise and containing in its interior zo and no other singularity of f(z). Particularly useful to us is the formula for evaluation of residues at pole singularities. If f(z) has a pole of order k at z= Zo, then the residue of f(z)at zo is given by (k-1)!z→z0 -k-1 (z-x0)f(x) (A.13) Cauchy-Goursat and residue theorems. It can be shown that if f(z) is analytic at all points on and within a simple closed contour C, then f(z)dz=0 This central result is known as the Cauchy-Goursat theorem. We shall not offer a proof, but shall proceed instead to derive a useful consequence known as the residue theorem 0 2001 by CRC Press LLCexpansion, a generalization of the Taylor expansion involving both positive and negative powers of z − z0: f (z) = ∞ n=−∞ an(z − z0) n = ∞ n=1 a−n (z − z0)n + ∞ n=0 an(z − z0) n. The numbers an are the coefficients of the Laurent expansion of f (z) at point z = z0. The first series on the right is the principal part of the Laurent expansion, and the second series is the regular part. The regular part is an ordinary power series, hence it converges in some disk |z−z0| < R where R ≥ 0. Putting ζ = 1/(z−z0), the principal part becomes ∞ n=1 a−nζ n; this power series converges for |ζ | < ρ where ρ ≥ 0, hence the principal part converges for |z − z0| > 1/ρ
r. When r < R, the Laurent expansion converges in the annulus r < |z − z0| < R; when r > R, it diverges everywhere in the complex plane. The function f (z) has an isolated singularity at point z0 if f (z) is not analytic at z0 but is analytic in the “punctured disk” 0 < |z − z0| < R for some R > 0. Isolated singularities are classified by reference to the Laurent expansion. Three types can arise: 1. Removable singularity. The point z0 is a removable singularity of f (z) if the principal part of the Laurent expansion of f (z) about z0 is identically zero (i.e., if an = 0 for n = −1, −2, −3,...). 2. Pole of order k. The point z0 is a pole of order k if the principal part of the Laurent expansion about z0 contains only finitely many terms that form a polynomial of degree k in (z − z0)−1. A pole of order 1is called a simple pole. 3. Essential singularity. The point z0 is an essential singularity of f (z) if the principal part of the Laurent expansion of f (z) about z0 contains infinitely many terms (i.e., if a−n = 0 for infinitely many n). The coefficient a−1 in the Laurent expansion of f (z) about an isolated singular point z0 is the residue of f (z) at z0. It can be shown that a−1 = 1 2π j   f (z) dz (A.12) where  is any simple closed curve oriented counterclockwise and containing in its interior z0 and no other singularity of f (z). Particularly useful to us is the formula for evaluation of residues at pole singularities. If f (z) has a pole of order k at z = z0, then the residue of f (z) at z0 is given by a−1 = 1 (k − 1)! lim z→z0 dk−1 dzk−1 [(z − z0) k f (z)]. (A.13) Cauchy–Goursat and residue theorems. It can be shown that if f (z) is analytic at all points on and within a simple closed contour C, then  C f (z) dz = 0. This central result is known as the Cauchy–Goursat theorem. We shall not offer a proof, but shall proceed instead to derive a useful consequence known as the residue theorem