operations on contours in the complex plane
DESCRIPTION
Contours can be added, composed and multiplied and various operations can be applied in a contour space.TRANSCRIPT
An Elementary Note: Operations on Contours in the Complex Plane
John Gill March 2015
l. Algebra of Contours (See [1])
Zeno (or equivalent parametric) contours are defined algorithmically as a distribution of points
{ }, 0lim
n
k n knz
=→∞ by the iterative procedure: ( ), 1, , 1, , kk n k n k n k n n
z z zη ϕ− −= + ⋅ which arises from the
following composition structure:
( ), , , 1, , , 1, 1,( ) lim ( ) , ( ) ( ) , ( ) ( , ) , ( ) ( )k
n n k n k n k n k n k n n nnn
G z G z G z g G z g z z z G z g zη ϕ−→∞
= = = + ⋅ = .
Usually ,
1k n
nη = , providing a partition of the unit time interval. ( )zγ is the continuous arc
from z to ( )G z that results as n→ ∞ (Euler’s Method is a special case of a Zeno contour).
When ( , )z tϕ is well-behaved an equivalent closed form of the contours , ( )z t , has the
property ( , )dz
z tdt
ϕ= , with vector field ( , ) ( , )f z t z t zϕ= + . Siamese contours are streamlines
or pathlines joined at their origin and arising from different vector fields. It will be assumed in
most of what follows that all contours in a particular discussion have identical initial points.
Contours will be abbreviated, using the iterative algorithm, as
( ), 1, , 1, 1, , 1, 1,: ( , ) ( , )k kk n k n k n k n k n k n k n k nn nz z z z f z zγ η ϕ η− − − − −= + = + −
Or : ( , )dz
z tdt
γ ϕ=
A parametric form of ( )zγ : ( )z z t= , exists when the equation ( , )dz
z tdt
ϕ= admits a closed
solution. For example 22
0: ( ) ( , ) (1 4 )t itz t z e z t z itγ ϕ+= ⇔ = + .
A Contour Sum 1 2γ γ γ= ⊕�
is defined as follows: 1 2
: ( , ) ( , ) ( , )dz
z t z t z tdt
γ ϕ ϕ ϕ= = +
For 1 1
: ( , )dz
z tdt
γ ϕ= and 2 2
: ( , )dz
z tdt
γ ϕ=
And a Scalar Product:
1
γ α γ= � : 1( , )
dzz t
dtα ϕ= ⋅ ,
Combined to show a distributive feature:
1 2 1 2( ) ( ) ( )γ α γ γ α γ α γ= ⊕ = ⊕� �
� � � : 1 2 1 2
( ( , ) ( , )) ( , ) ( , )dz
z t z t z t z tdt
α ϕ ϕ αϕ αϕ= + = +
A Contour Product is defined: 1 2γ γ γ= ⊗�
: 1 2( , ) ( , )
dzz t z t
dtϕ ϕ= ⋅ , from which one derives
1 2 2 1( ) ( )γ α γ γ α γ γ= ⊗ = ⊗� �
� �
Define Contour Composition: 1 2 1, , , 1 2: ( ( , ), )k n k n k nz z z t tγ γ γ η ϕ ϕ+= = + ⋅� or 1 2
dz
dtϕ ϕ= �
A norm γ of a contour in a 0z - based space can be formulated as 0
[0,1]
( , )t
Sup z tγ ϕ∈
= ,
giving rise to a metric: ( )1 2 1 2 1 2, ( 1)d γ γ γ γ γ γ= − = ⊕ − ⋅
�.
ll. Operations on Contours
Set : 1I
dz
dtγ = , the identity contour
0( )z t z t= + .
Define a linear operator
( ) : ( , )dz
T z tdt
γ α γ α ϕ= ⋅ = ⋅ so that 1 2 1 2
( ) ( ) ( )T T Tγ γ γ γ⊕ = ⊕�� ��
If we assume ( )zϕ is analytic the following are linear operators as well:
: ( )dz
zdt
γ ϕ=∫ ∫ (in the sense of antiderivative)
and : ( )z
dzD D z
dtγ ϕ=
Example 1 : : ( ) : ( )dz dz
Cos z Sin zdt dt
γ γ= ⇒ =∫ . .5 3z i= − + , γ green, γ∫ red.
Contour exponentiation: : ( ) : ( )dz dz
z zdt dt
α αγ ϕ γ ϕ= ⇒ =
Example 2 : 2 1 2(1 ): : i idz dzz z
dt dtγ γ + += ⇒ = ,
01.5 1.5z i= −
Orthogonality of contours: : ( )dz
zdt
γ ϕ= , * : ( )
dzi z
dtγ ϕ= −
* γ γ⇒ ⊥
Example 3 : : ( ) ( )dz
xCos x y iySin x ydt
γ ϕ= = + + −
Consider a contour , 1, , 1 1,
: ( )k n k n k n k nz z zγ µ ϕ− −= + , and a second function
2( )zϕ .
Define an Integration Operator 2 , 1, , 2 1, 1 1,
: ( ) ( )k n k n k n k n k n
z zγϕ ζ ζ µ ϕ ϕ− − −= + . Then
1
, 0, , 1, 2 1, , 1, 2 2 1
1 1 ( ) 0
( ) ( )( ) ( ) ( ( )) ( ( ))n n
n n n k n k n k n k n k n
k k z
z z z z dz z t z t dtγ
ζ ζ ζ ζ ϕ ϕ ϕ ϕ− − −= =
− = − = − → =∑ ∑ ∫ ∫
The result: The integral of the second function along the contour described by the first.
This is analogous to 1 2 1( ) and ( ) ( )
dz dz z z
dt dt
ζϕ ϕ ϕ= = ⋅
A more complete description can be found in [2]. The vector , 0,n n nζ ζ−����������
provides a graphical depiction of this
integral.
Define a Composition Operator 2 , 1, , 2 1 1,
: ( )k n k n k n k n
zϕ γ ζ ζ µ ϕ ϕ− −= +� � .
An argument similar to the previous leads to
1 1
, 0, 2 , , , 2 2 1
1 0 0
( / ) ( / ) ( ( ( )))n
n n n k n k n k n
k
z dz dt dt z t dtζ ζ ϕ µ µ ϕ ϕ ϕ=
− = ∆ → =∑ ∫ ∫ .
Analogous to 1 2 1( ) and ( ( ))
dz dz z
dt dt
ζϕ ϕ ϕ= =
Define a Contour Mixing Operation
( )1 2 , 1, 1 1, , 1, 2 1,, : ( ) and ( )
k n k n n k n k n k n n k nz z zγ γ µ ϕ ζ ζ ζ µ ϕ− − − −= + = +M ,
analogous to the system
1( )
dz
dtϕ ζ= and
2( )
dz
dt
ζϕ= , , z x iy iζ τ υ= + = +
Example 4 : ( ) ( )( ) ( ) ( ) ( )dz
Sin Cos i Cos Sindt
τ τ υ υ τ υ τ τ υ υ τ υ= + + − + + − − and
( ) ( )d
xCos y iySin xdt
ζ= + ,
1
0
( ) ( , ) ( , )z t t dtν ω ω ζ ω= ∫M, (0) (0)zω ζ= =
Topographical image of the ω -plane:
Further mixing of contours:
1 , 1, 1 1, 2 , 1, 2 1, 1,: ( ) and : ( )
k n k n n k n k n k n n k n k nz z z zγ µ ϕ γ ζ ζ µ ϕ ζ− − − − −= + = +
Expanding 2
γ two different ways:
(1) ( ) ( )1 1, 0 2 1, 1, 0 2
1 1
1 1( ) ( )
n n
k kn n k n k n n n
k k
z zn n
ζ ζ ϕ ζ ζ ϕ ζ− −− −
= =
= + = +∑ ∑
(2) ( )1, 0 2
1
11 ( )
n
kn n n
k
zn
ζ ζ ϕ −
=
= +
∏
Using the notation of virtual integral, (1) becomes
1
, 0 2
0
( , ) ( )n n z t t dtζ ζ ψ ζ≈ + ∫
And simple manipulations produce the following form of (2)
1
2
0
( , )
, 0
z t dt
n ne
ψ
ζ ζ∫
�
Combined, these two expressions give
1
2
0
1 ( , )
2
0
( ) ( , ) ( ) 1t dt
t t dt eψ α
λ α ψ α ζ α
∫ = −
∫ �
Observe that the contour along which integrals are evaluated is 1
γ .
This is analogous to
2
0
( ( ))
2 1 0
( )( ( )) ( ) , ( ) ( )
t
z t dtd t dzz t t z t
dt dte
ϕζϕ ζ ϕ ζ ζ
∫= ⋅ = ⇒ =
Recall, under ideal conditions: ( , ) ( ( )) , (0)t z t zψ α ϕ α= = .
A general scenario: 2 1( , ) , ( )
d dzz z
dt dt
ζϕ ζ ϕ= =
Example 5a : 2
1 2
1, ( ) ( )
10z xCos x y iySin x yϕ ϕ= = + + − , α - plane
1
2 1 0
0
( ) exp ( , ) , ( ) , dz
t dt z zdt
λ α α ψ α ϕ α
= ⋅ = = ∫
Example 5b : 2 1( ( )) , ( )
d dzz t z
dt dt
ζϕ ϕ= = ⇒
1
2 2 2
0
( ) ( , ) , ( , ) ( ( ))t dt t z tλ α ψ α ψ α ϕ= =∫
[1] J. Gill, A Space of Siamese Contours in Time-dependent Complex Vector Fields, Comm. Anal. Th. Cont. Frac., Vol XX (2014)
[2] J. Gill, A Note on Integrals & Hybrid Contours in the Complex Plane, Comm. Anal. Th. Cont. Frac., Vol XX (2014)