Week 7 2. The Laurent series and the Residue Theorem Theorem 1: Laurent’s Theorem If f(z) is analytic in an annulus (i.e. a domain between two concentric circles) with centre z0, then f(z) can be represented by the Laurent series, (1) Proof: Kreyszig, section 16.1 (non-examinable) ۞ The second term on the r.-h.s. of (1) is called the principal part of the Laurent series.

Comment: Instead of (1), one can write (observe the lower limit of summation). Example 1: Let’s find the principal part of the Laurent series of f(z) = z –3 e z at z = 0: Hence, b3 = b2 = 1, b1 = ½.

tan z has an isolated singularities at z = ±π/2, ±3π/2, ±5π/2... ۞ We say that a function f(z) has a singularity at z = z0 if f(z) is not analytic (perhaps not even defined) at z0, but every neighbourhood of z0 contains points where f(z) is analytic. ۞ We say that a function f(z) has an isolated singularity at z = z0 if f(z) has a singularity at z0, but is analytic in a neighbourhood of z0 (not including z0). Example 2: tan z has an isolated singularities at z = ±π/2, ±3π/2, ±5π/2... tan z –1 has a non-isolated singularity at z = 0 (and also isolated singularities at z = ±2/π, ±2/(3π), ±2/(5π)...). Comment: A function with an isolated singularity at z0 always has a Laurent series at z0 (why?).

۞ If the Laurent series of a function f(z) at z = z0 has a finite principal part (i.e. bn = 0 for all n > N), and if bN ≠ 0, we say that f(z) has at z0 a pole of order N. Example 3: Determine the order of the pole of the function from Example 1. ۞ If the principal part of the Laurent series of f(z) is infinite, we say that f(z) has an essential singularity at z0. Example 4: Show that e1/z has an essential singularity at z = 0.

Example 5: Behaviour of functions near poles and ESs Find out whether the following functions have a limit at z = 0 when this point is approached along the positive (negative) part of the real (imaginary) axis: Comment: A function with a branch point at z = z0 doesn’t have a Laurent series at z = z0 (explain why Theorem 1 doesn’t hold in this case). Thus, branch points are neither poles, nor essential singularities.

۞ A function f(z) is said to be analytic at infinity if g(z) = f(1/z) is analytic at z = 0. Example 6: Are the following functions: analytic at infinity? If they are not, determine the type of their singularity there. Theorem 2: Let a function f(z) be analytic on the extended complex plane (i.e. the complex plane + infinity). Then, f(z) = const. Proof: This theorem follows from Liouville’s Theorem (Theorem 5.5).

۞ The coefficient b1 of the Laurent series of a function f(z) at z = z0 is called the residue of f(z) at z0 and is denoted by Useful formulae: Let f(z) be analytic at z = z0. Then and, in general,

Example 7:

Theorem 3: the Residue Theorem Let f(z) be analytic at all points of a simply connected domain D except finitely many poles or essential singularities located at zn (where n = 1, 2... N). Let also f(z) be analytic on C, where the contour C is the boundary of D. Then where C is positively oriented (i.e. traversed in the counter- clockwise direction). Proof: This theorem follows from the principle of deformation of the path and Example 12 from TS 2, where we showed that...

where C is a positively oriented circle centred at z = z0. ۞ A function analytic in a domain D is often called holomorphic in D. ۞ A function that is analytic in a domain D except finitely many poles is often called meromorphic in D. Example 8: Calculate where C is a positively oriented unit circle centred at z = 0.