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)