Examples of analytic functions
ez + e
cosh z =
2
eiz + e
cos z =
2
z
iz
ez + e
, sinh z =
2
, sinh z =
eiz
e
2i
z
iz
ln z can be defined as inverse function of ez
eln z = z
i.e. by the requirement
z+2⇡i
= ez
BUT e
so the logarithm is not uniquely defined
if z = rei✓ then ln z = ln r + i✓ = ln |z| + i✓
so a possible unique definition is Logz = ln r + iArgz
Logz is discontinuous along the negative real axis, analytic in the rest of C
1
2
z =e
1
2 Logz
so z
1
2
is defined on the same domain as Logz
Laurent series
f (z) =
1
X
cn (z
z0 ) n
n= 1
If this series converges absolutely of a disk with excluded centre z0
then it defines an analytic function with a singularity at z0
Example
f (z) =
1
1
z
has a singularity at 1
Problem: find the Laurent series of f (z) =
z2
1
3z + 2
at 1
Differentiation of power and Laurent series
Power or Laurent series can be differentiated term by term
Given a Laurent series
Then
f 0 (z) =
2
X
n= 1
f (z) =
1
X
cn (z
z0 ) n
n= 1
(n + 1)cn+1 z n +
1
X
(n + 1)cn+1 z n
n=0
0
z
z
(e
)
=
e
Show using power series expansion that
1
X
1
( 1)n+1 n
0
z show that f (z) =
Given f (z) =
1+z
n
n=1
What is f ?
We
begin
with
the
Note
that
curve
in the
y -plane
be represented
We
begin
the
Note
that
anyany
curve
in the
x, y x,-plane
can can
be represented
denotes
anwith
angle
orterminology.
aterminology.
length.
in terms
termsofofaasingle
singleparameter;
parameter;this
this
is
often
indicated
by
t,
independent
of
whether
it
in
is
often
indicated
by
t,
independent
of
whether
it
Contour integration
denotes
anangle
angle
ora alength.
length.
Closed an
Curve
(ororContour)
denotes
A contour C, represented by the function z
Closed
Curve
Closed
(orContour)
Contour)
f
fCurve
. (or
f t on the interval I :
≤ t ≤ , is closed if
A
function
z z f t f on
thethe
interval
I : I :≤ t ≤ t ,≤is closed
if if
A contour
contourC,
C,represented
representedbybythethe
function
t on
interval
, is closed
ff SimpleffClosed
. . Curve
A simple closed curve is one that does not cut itself. Formally, on the interval
I : ≤Closed
t ≤ , Curve
we
cannot have f t 1
f t 2 except when t 1
and t 2
.
Simple
Simple
Closed
Curve
A
simple
closed
curve
isisone
that
does
notnot
cutcut
itself.
Formally,
on the
interval
ACircles
simpleand
closed
curve
one
that
does
itself.
Formally,
on
the interval
polygons are examples of simple closed curves, so that the parametric
II :: ≤≤ tt ≤≤ , ,we
when
t 1 t andand
t2 t . .
wecannot
cannothave
havef tf1 t f tf2 t except
except
when
2 of t but f ′ t can 1have discontinuities
2
representation z f t is a continuous1 function
The
direction of traverse of a closed curve is an important quantity. It is taken to be positive if
Circles and polygons are examples of simple closed curves, so that the parametric
Circles
polygons are direction
examples
simple ifclosed
curves,
so
that theThis
parametric
taken inand
an anti-clockwise
andofnegative
in a clockwise
direction.
is the
′
The
function
z
=
f
(t)
is
continuous
and
the
derivative
is
piecewise
continuous.
representation
z asf employed
t is a continuous
function of t of
but fin polar
t ′ canco-ordinates.
have discontinuities The
same
convention
with
the
measurement
representation z f t is a continuous function of t but f t can have discontinuities The
direction of traverse of a closed curve is an important quantity. It is taken to be positive if
direction of traverse of a closed curve is an important quantity. It is taken to be positive if
taken
in an anti-clockwise
Simply-Connected
Domaindirection and negative if in a clockwise direction. This is the
taken in an anti-clockwise direction and negative if in a clockwise direction. This is the
same
convention as employed
with
the
measurement
ofdoesinnot
polar
co-ordinates.
A
simply-connected
domain
D
in
the
z-plane
is
one
that
contain
any holes.
same convention as employed with the measurement of in polar co-ordinates.
Simply-Connected Domain
Generally we will restrict
attention to simple closed curves on simply-connected domains.
Simply-Connected
Domain
A simply-connected domain D in the z-plane is one that does not contain any holes.
A2.3.1
simply-connected
domain D in the z-plane is one that does not contain any holes.
The Integral Theorems
Generally
we will
restrictareattention
simpleproof.
closedThe
curves
on simply-connected
The important
theorems
all statedtowithout
notation
indicates that thedomains.