1

2. The Laurent series and the Residue Theorem

Week 7

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,

Theorem 1: Laurent’s Theorem

.)(

)()(1 00

0

nn

n

n

nn zz

bzzazf

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.

(1)

2

(observe the lower limit of summation).

n

nn zzczf )()( 0

Example 1:

Comment:

Instead of (1), one can write

Let’s find the principal part of the Laurent series of f(z) = z –

3 e z at z = 0:

.62

11

!

1)(

32

30

3

zzz

zn

z

zzf

n

n

Hence, b3 = b2 = 1, b1 = ½.

3

۞ 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?).

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۞ 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.

۞ 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 3:

Determine the order of the pole of the function from Example 1.

5

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:

Example 5: Behaviour of functions near poles and ESs

).(cos)b(,)a( 12 zz

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.

Comment:

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.

13 )e(,ln)d(,e)c(,cos)b(,)a( zzzz z

Theorem 2:

Let a function f(z) be analytic on the extended complex plane (i.e. the complex plane + infinity).

Proof:

This theorem follows from Liouville’s Theorem (Theorem 5.5).

Then, f(z) = const.

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۞ 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

.),(res 01 zzfb

Useful formulae:

Let f(z) be analytic at z = z0. Then

),(,)(

)(res),(,

)(res 002

000

0

zfzzz

zfzfz

zz

zf

and, in general,

.d

d

)!1(

1,

)(

)(res

0

1

1

00 zz

n

n

n z

f

nz

zz

zf

8

Example 7:

.11,1

res)d(,10,1

res)c(

,10,)(

sinres)b(,10,

sinres)a(

22

z

z

z

z

z

z

z

z

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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.

Theorem 3: the Residue Theorem

Proof:

This theorem follows from the principle of deformation of the path and Example 12 from TS 2, where we showed that...

Then

,]),([resi2d)(1

N

nnC

zzfzzf

where C is positively oriented (i.e. traversed in the counter-clockwise direction).

10

,1 ifi2

,1 if0d)( 0 n

nzzz

C

n

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

,dsin

)b(dsin

)a(32 CC

zz

zz

z

z

where C is a positively oriented unit circle centred at z = 0.