1 2. the laurent series and the residue theorem week 7 if f(z) is analytic in an annulus (i.e. a...
TRANSCRIPT
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:
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۞ 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.