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Complex number:
Complex algebra:
Complex conjugation:
2
.)1 real, are and (both iyxiyx z
Cauchy-Riemann conditions
Complex algebra
2112212121221121
2121221121
vectors.)2d to(Anologous
zzicycxiyxcczyxyxiyyxxiyxiyxzz
yyixxiyxiyxzz
iyxiyx z ** )(22* ))(( yxiyxiyx zz
Modulus (magnitude): 212122* zzzz yxrzzz
Polar representation: ireiriyx z )sin(cos
Argument (phase):
)arg()arg()arg(
)quadrants. 3rdor 2nd in the is if ( arctan)arg(
2121 zzzz
zx
yz
3
Functions of a complex variable:
All elementary functions of real variables may be extended into the complex plane.
0
2
0
2
!!2!11
!!2!11 :Example
n
nz
n
nx
n
zzze
n
xxxe
A complex function can be resolved into its real part and imaginary part:
2222
2222
11
2)()( :Examples
),(),()(
yx
yi
yx
x
iyxz
xyiyxiyxz
yxivyxuzf
Multi-valued functions and branch cuts:
ivunirrereznii )2(ln]ln[)ln(ln :1 Example )2(
To remove the ambiguity, we can limit all phases to (-,). = - is the branch cut. lnz with n = 0 is the principle value.
2/)2(2121)2(2121 )( :2 Example niniierrerez
We can let z move on 2 Riemann sheets so that is single valued everywhere. 21)()( irezf
4
Analytic functions: If f (z) is differentiable at z = z0 and within the neighborhood of z=z0, f (z) is said to be analytic at z = z0. A function that is analytic in the whole complex plane is called an entire function.
Cauchy-Riemann conditions for differentiability
Cauchy-Riemann conditions
z
zf
z
zfzzf
dz
dfzf
zz
)(lim
)()(lim)('
00
In order to let f be differentiable, f '(z) must be the same in any direction of z.
Particularly , it is necessary that
x
v
y
u
y
v
x
u
y
v
y
ui
yi
viuzfyiz
x
vi
x
u
x
viuzfxz
y
x
,
have we themEquating
.lim)(' ,For
.lim)(' ,For
0
0
Cauchy-Riemann conditions x
v
y
u
y
v
x
u
,
5
Conversely, if the Cauchy-Riemann conditions are satisfied, f (z) is differentiable:
.
1 and ,lim
limlim)(
lim
0
000
y
vi
y
u
ix
vi
x
u
yix
yixx
vi
x
u
yix
yx
ui
x
vx
x
vi
x
u
yix
yy
vi
y
ux
x
vi
x
u
z
zf
dz
df
z
zzz
More about Cauchy-Riemann conditions:
1) It is a very strong restraint to functions of a complex variable. 2) 3)
4)
2100 cvcuvuvuy
v
y
u
x
v
x
u
.)()( iy
vi
iy
u
y
ui
y
v
x
vi
x
u
dz
df
analytic.not but continuous everywhere is ,e.g.
0002
1
2
1
:on dependsonly )( that so ,0 toEquivalent
***
*
iyxf
y
vi
y
ui
x
vi
x
u
y
fi
x
f
iy
f
x
f
z
y
y
f
z
x
x
f
z
f
zz,zfz
f *
6
Reading: General search for Cauchy-Riemann conditions: Our Cauchy-Riemann conditions were derived by requiring f '(z) be the same when z changes along x or y directions. How about other directions? Here I do a general search for the conditions of differentiability.
x
v
y
u
zf
y
v
x
u
ip
x
u
y
vi
x
v
y
u
ip
py
v
x
vp
y
u
x
uiip
y
vi
y
u
ip
py
v
x
vip
y
u
x
u
dp
d
dp
zdf
pzf
zpdx
dy
dx
dyi
dx
dy
y
v
x
vi
dx
dy
y
u
x
u
idydx
dyy
vdx
x
vidy
y
udx
x
u
idydx
idvdu
dz
dfzf
.conditionsRiemann -Cauchy same get the we
,directions allat same thebe )(' require weif is,That
1
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10
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hunder whiccondition thefind want to We. of change theofdirection the,let Now
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2
7
Cauchy’s theorem
Cauchy’s integral theorem
Contour integral: Cauchy’s integral theorem: If f (z) is analytic in a simply connected region R, [and f ′(z) is continuous throughout this region, ] then for any closed path C in R, the contour
integral of f (z) around C is zero:
CCC
z
zudyvdxivdyudxidydxivudzzf )()())(()(
2
1
C
dzzf 0)(
Proof using Stokes’ theorem: σVλV ddSC
dxdyy
V
x
VdyVdxV
S
xy
Cyx
0
)()()(
dxdy
y
v
x
uidxdy
y
u
x
v
udyvdxivdyudxdzzf
SS
CCC
8
Contour deformation theorem: A contour of a complex integral can be arbitrarily deformed through an analytic region without changing the integral.
1) It applies to both open and closed contours. 2) One can even split closed contours. Proof: Deform the contour bit by bit.
Examples: 1) Cauchy’s integral theorem. (Let the contour shrink to a point.) 2) Cauchy’s integral formula. (Let the contour shrink to a small circle.)
Cauchy-Goursat proof: The continuity of f '(z) is not necessary.
Corollary: An open contour integral for an analytic function is independent of the path, if there is no singular points between the paths.
1
2
2
1
)()()()( 12
z
z
z
zdzzfzFzFdzzf
z1
z2
z1
z2 C2
C1
C1
C2
C1
C2 C3
“nails”
“rubber bands”
9
Cauchy’s integral formula
Cauchy’s integral formula: If f (z) is analytic within and on a closed contour C, then for any point z0 within C,
)(2
)0(Let )()()(
0)()()()(
:Proof
)(
2
1)(
0
0
2
0
00
0000
0
0
0
201
zif
rdriere
rezfdz
zz
zfdz
zz
zf
dzzz
zfdz
zz
zfdz
zz
zfdz
zz
zf
dzzz
zf
izf
i
i
i
CC
LCLC
C
C C0
L1
L2
z0
Can directly use the contour deformation theorem.
10
Derivatives of f (z):
C n
ndz
zz
zf
i
nzf
1
0
0)( )(
2
!)(
Corollary: If a function is analytic, then its derivatives of all orders exist. Corollary: If a function is analytic, then it can be expanded in Taylor series.
Cauchy’s inequality: If is analytic and bounded,
then
n
nzazf )( ,)(||
Mzfrz
bounded.) is is,(That . n
n
n aMra
Mrar
Mdz
z
zfadz
z
zf
i
nanf
n
nnrz nnrz nn
n || 1|| 1
)( )(
2
1)(
2
!!)0( :Proof
Liouville’s theorem: If a function is analytic and bounded in the entire complex plane, then this function is a constant.
.)( 0.for 0 then ,let , :Proof 0azfnarr
Ma nnn
Fundamental theorem of algebra: has n roots. Suppose P(z) has no roots, then 1/P(z) is analytic and bounded as Then P(z) is constant. That is nonsense. Therefore P(z) has at least one root we can divide out.
)0,0( )(0
n
n
i
i
i anzazP
.z
11
Morera’s theorem: If f (z) is continuous and for every closed contour within a simply connected region, then f (z) is analytic in this region.
C
dzzf 0)(
analytic is )()('
analytic is )(
)()(')()()(0)(
:Proof
12
2
1
zfzF
zF
zfzFzFzFdzzfdzzfz
zC
)()(),0(),0(
),0()0,(),(
then),,()(Let
?)()()(Why
2121
2121
21
12
2
1
2
1
zFzFzGzG
zGzGzzG
zzGdzzf
zFzFdzzf
z
z
z
z
12
Analytic continuation
Laurent expansion
Taylor expansion for functions of a complex variable: Expanding an analytic function f (z) about z = z0, where z1 is the nearest singular point.
00
0)(
01
0
00
10
0
0
0 0
0
0
00
00
!
)(
')'(
)'(
2
1'
)'(
)'(
2
1
')'(
)'('
2
1'
'1)'(
)'(
2
1
')()'(
)'(
2
1'
'
)'(
2
1)(
n
nn
nC n
n
Cn
n
n
C
n
n
C
CC
zzn
zf
dzzz
zfzz
idz
zz
zfzz
i
dzzz
zfzz
zz
idz
zz
zzzz
zf
i
dzzzzz
zf
idz
zz
zf
izf
z
13
Schwarz’s reflection principle: If f (z) is 1) analytic over a region including the real axis, and 2) real when z is real, then
).()( **zfzf
)()(
!
)()( :Proof
**
00
0)(
zfzf
xzn
xfzf
n
nn
Examples: most of the elementary functions.
Analytic continuation: Suppose f (z) is analytic around z = z0, we can expand it about z = z0 in a Taylor series: This series converges inside a circle with a radius of convergence , where a0 is the nearest singularity from z = z0. We can also expand f (z) about another point z = z1 within the circle R0: . In general, the new circle has a radius of convergence and contains points not within the first circle.
14
0
00
)(
!
)()(
m
mm
zzm
zfzf
000 zR a
0
11
)(
!
)()(
n
nn
zzn
zfzf
z0
a0
z1
R0
R1 a1
nmn
nnmm
nm
nmm
n
zznmn
zzzfzf
zznm
zfzf
,01
010)(
010
)(
1)(
)!(!
)()( expansion, second theinto Plug
)!(
)()( expansion,first theFrom
111 zR a
Consequences: 1) f (z) can be analytically continued over the complex plane, excluding singularities. 2) If f (z) is analytic, its values at one region determines its values everywhere.
15
Laurent expansion
Laurent expansion
Problem: Expanding a function f (z) that is analytic in an annular region (between r and R).
nC n
n
nC n
n
nC n
n
mC
m
mn
C n
n
mC
m
mn
C n
n
CC
CC
CC
LCLC
zz
dzzfzz
i
zz
dzzfzz
izz
dzzfzz
i
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dzzfzz
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izzzz
dzzf
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izz
dzzf
i
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izf
10
0
11
0
00
10
0
1
10
001
0
0
001
001
0
0
0
00
0
00
0000
~
)'(
')'()(
2
1
)'(
')'()(
2
1
)'(
')'()(
2
1
')'()'()(
1
2
1
)'(
')'()(
2
1
')'()'()(
1
2
1
)'(
')'()(
2
1
'1)(
')'(
2
1
'1)'(
')'(
2
1
)()'(
')'(
2
1
)()'(
')'(
2
1
'
')'(
2
1
'
')'(
2
1
'
')'(
2
1)(
21
21
21
21
21
21
2211
L1
L2
C is any contour that encloses z0 and lies between r and R (deformation theorem).
16
C nn
n
n
nzz
dzzf
iazzazf
10
0)'(
')'(
2
1 ,)()(
Laurent expansion:
1) Singular points of the integrand. For n < 0, the singular points are determined by f (z). For n ≥0, the singular points are determined by both f (z) and 1/(z'-z0)
n+1. 2) If f (z) is analytic inside C, then the Laurent series reduces to a Taylor series:
3) Although an has a general contour integral form, In most times we need to use straight forward complex algebra to find an.
.0 ,0
,0 ,!
)( 0)(
n
nn
zf
a
n
n
17
Laurent expansion: Examples
Example 1: Expand about z0=1. 2
3
1)(
z
zzf
)1(3
1
3
1
1
1
1)1(3)1(3)1(
1
]1)1[(
122
23
2
3
2
3
zzzz
zzz
z
z
z
z
Example 2: Expand about z0=i.
)(84
11
2
)(2
1
2
11
2
1
21
1
2
11
2
1
2
11
2
111
2
1
1
1)(
02
2
izi
iz
i
iziiizi
i
iziizi
iziiziizizizzf
n
n
n
1
1)(
2
zzf
18
Branch points and branch cuts
Singularities
Poles: In a Laurent expansion
then z0 is said to be a pole of order n. A pole of order 1 is called a simple pole. A pole of infinite order (when expanded about z0) is called an essential singularity.
The behavior of a function f (z) at infinity is defined using the behavior of f (1/t) at t = 0.
Examples:
,0 and 0for 0 if ,)()( 0
nm
m
m
m anmazzazf
1200
12 1
)!12(
)1(1sin ,
)!12(
)1(sin )2
nn
n
n
nn
tntn
zz
.at pole single a has 22
14
11
2
1
2/)(1
1
4
11
2
1
)(2
11
2
111
2
1
))((
1
1
1)1
2
2
izi
iz
i
iz
izi
iiziziiziiziiziziizizz
sinz thus has an essential singularity at infinity.
infinity.at 2order of pole a has 1 )3 2 z
19
Branch points and branch cuts:
Branch point: A point z0 around which a function f (z) is discontinuous after going a small circuit. E.g.,
Branch cut: A curve drawn in the complex plane such that if a path is not allowed to cross this curve, a multi-valued function along the path will be single valued. Branch cuts are usually taken between pairs of branch points. E.g., for , the curve connects z=1 and z = can serve as a branch cut.
.lnfor 0 ,1for 1 00 zzz-z
1 z-
Examples of branch points and branch cuts:
)sin(cos)( .1 aiarzzfaa
If a is a rational number, then circling the branch point z = 0 q times will bring f (z) back to its original value. This branch point is said to be algebraic, and q is called the order of the branch point. If a is an irrational number, there will be no number of turns that can bring f (z) back to its original value. The branch point is said to be logarithmic.
,/ qpa
20
)1)(1()( .2 zzzf
We can choose a branch cut from z = -1 to z = 1 (or any curve connecting these two points). The function will be single-valued, because both points will be circled.
Alternatively, we can choose a branch cut which connects each branch point to infinity. The function will be single-valued, because neither points will be circled.
It is notable that these two choices result in different functions. E.g., if , then iif 2)(
choice. second for the
2)( and choicefirst for the 2)( iifiif
A B
A B
21
Mapping
Mapping
Mapping: A complex function can be thought of as describing a mapping from the complex z-plane into the complex w-plane. In general, a point in the z-plane is mapped into a point in the w-plane. A curve in the z-plane is mapped into a curve in the w-plane. An area in the z-plane is mapped into an area in the w-plane.
),(),()( yxivyxuzw
z0
z1
z
y
x
w0
w1
w
v
u
Examples of mapping:
Translation:
0 zzw
Rotation:
0
0
0
0
0
or ,
rrerree
zzw
iii
22
Inversion:
r
ree
zw
i
i
11
or ,1
In Cartesian coordinates:
.,11
22
22
22
22
vu
vy
vu
ux
yx
yv
yx
xu
iyxivu
zw
A straight line is mapped into a circle:
.0)( 22
2222
vauvub
bvu
au
vu
vbax y
23
Conformal mapping
Conformal mapping: The function w(z) is said to be conformal at z0 if it preserves the angle between any two curves through z0.
If w(z) is analytic and w'(z0)0, then w(z) is conformal at z0.
Proof: Since w(z) is analytic and w'(z0)0, we can expand w(z) around z = z0 in a Taylor series:
.)(
).)(('or ),('lim
))((''2
1))((')(
121200
0000
0
0
0
200000
0
aa
zzAeww
zzzwwwzwzz
ww
zzzwzzzwzww
i
zz
1) At any point where w(z) is conformal, the mapping consists of a rotation and a dilation.
2) The local amount of rotation and dilation varies from point to point. Therefore a straight line is usually mapped into a curve.
3) A curvilinear orthogonal coordinate system is mapped to another curvilinear orthogonal coordinate system .