1. 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.

2. 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.

3. 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?).

4. ۞ 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.

5. 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.

6. ۞ 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).

7. ۞ 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,

8. Example 7:

9. 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...

10. 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.