complex number : complex algebra: complex conjugation:
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Complex number:Complex algebra:
Complex conjugation:
1
.)1 real, are and (both iyxiyx z
October 3 Cauchy-Riemann conditions
6.1 Complex algebra
Chapter 6 Functions of a Complex Variable I
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vectors.)2d to(Anologous
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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
2
Functions of a complex variable:All elementary functions of real variables may be extended into the complex plane.
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2
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n
nz
n
nx
n
zzze
n
xxxe
A complex function can be resolved into its real part and imaginary part:
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2222
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2)()( :Examples
),(),()(
yx
yi
yx
x
iyxz
xyiyxiyxz
yxivyxuzf
Multi-valued functions and branch cuts:
ivunirrerez nii )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.
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We can let z move on 2 Riemann sheets so that is single valued everywhere.21)()( irezf
3
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
6.2 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
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Cauchy-Riemann conditions x
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Conversely, if the Cauchy-Riemann conditions are satisfied, f (z) is differentiable:
.
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limlim)(
lim
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More about Cauchy-Riemann conditions:
1) It is a very strong restraint to functions of a complex variable.
2)
3)
4)
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5
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.
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6
Read: Chapter 6: 1-2Homework: 6.1.7,6.1.17,6.2.1,6.2.8Due: October 14
7
October 7 Cauchy’s theorem
6.3 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
C yx
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
z2C2
C1
C1
C2
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C2 C3
“nails”
“rubber bands”
9
6.4 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
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2
1)(
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Can directly use the contour deformation theorem.
10
Derivatives of f (z):
C n
n dzzz
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
nn zazf )( ,)(
||Mzf
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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.
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n
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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.
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)()(')()()(0)(
:Proof
12
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12
Read: Chapter 6: 3-4Homework: 6.4.1,6.4.3,6.4.4Due: October 14
13
October 10 Analytic continuation
6.5 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.
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14
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 ofconvergence , where 0 is the nearestsingularity 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.
15
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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.
16
Read: Chapter 6: 5Homework: 6.5.2,6.5.5Due: October 21
17
October 12 Laurent expansion
6.5 Laurent expansionProblem: Expanding a function f (z) that is analytic in an annular region (between r and R).
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C is any contour that encloses z0 and lies between r and R (deformation theorem).
18
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nn zz
dzzf
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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.
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n
nn
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n
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19
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[(
1 22
23
2
3
2
3
zzzz
zzz
z
z
z
z
Example 2: Expand about z0=i.
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11
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1
2
11
2
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2
11
2
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20
Read: Chapter 6: 5Homework: 6.5.8,6.5.9,6.5.10Due: October 21
21
October 17 Branch points and branch cuts
6.6 SingularitiesPoles: 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
mm anmazzazf
1200
12 1
)!12(
)1(1sin ,
)!12(
)1(sin )2
nn
n
n
nn
tntn
zz
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14
11
2
1
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1
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11
2
1
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11
2
111
2
1
))((
1
1
1)1
2
2
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sinz thus has an essential singularity at infinity.
infinity.at 2order of pole a has 1 )3 2 z
22
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 aiarzzf aa
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
23
)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
AB
AB
24
Read: Chapter 6: 6Homework: 6.6.1,6.6.2,6.6.3Due: October 28
25
October 19 Mapping
6.7 MappingMapping: 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
00
0
0
or ,
rrerree
zzw
iii
26
Inversion:
rre
e
zw
ii
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
27
6.8 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))((')(
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0
0
200000
0
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 .
28
What happens if w'(z0) = 0?
Suppose w (n)(z0) is the first non-vanishing derivative at z0.
nn
BrreBe
nezz
n
zwww
n
niiinn
!)(!
1)(
!
)(0
0)(
0
This means that at z = z0 the angle between any two curves is magnified by a factor n and then rotated by .
29
Read: Chapter 6: 7-8Homework: 6.7.1,6.7.2,6.8.1Due: October 28
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