An Elementary Note: Contour & Virtual Integrals of Implicit Complex Functions - Images

John Gill July 2015

Consider the following problem: Evaluate ( )I z dzγ

ω= ∫ where both ( )z t and ( )zω are

defined implicitly, and :0 1t → . Assume the functions here are differentiable with respect to t

and (1) ( ), 0z t =F and (2) ( ) ( )f zω =G .

Solve for ( )0,0 0z =F and

0 0( ) ( )f zω =G using traditional technique.

Differentiate both expressions with respect to t , producing

(1a) ( , )z

dzz t

dtϕ= and (2a) ( ), ,

dz t

dtω

ωϕ ω= . Now convert to discrete formats:

(1b) ( )1, 1, 1,

, kk n k n n z k n nz z zη ϕ −

− −= + ⋅ and (2b) ( )1, 1, 1, 1,, , kk n k n n k n k n n

zωω ω η ϕ ω −− − −= + ⋅ , 1 k n≤ ≤ .

Define (3) ( ), ,

1

1( , ) ,

n

kn n k n z k n nn

k

S zn

η ω ϕ η=

= ⋅ ⋅ =∑ , Thus limn

nI S

→∞= .

In a force field : ( )F f f z= the work done along a contour γ is given by

1

0

Re ( ( )) '( )W f dz f z t z t dtγ

= =∫ ∫�

With the formula holding when both ( )f z and ( )z t are given implicitly.

The Zeno contours , 1, 1, 1,

( , )kk n k n n z k n nz z zη ϕ −

− −= + ⋅ , where ( , )z

d d dzz t

dt dt dt

ζϕ= ⇒ =

F, can be

expanded linearly:

1

0 1 1, 0 0 0 0 0 0

0

( , ) ( , ) ( , ) ( , )nn n n n nn n nz z z z z z z t dtη ψ η ψ η ψ ψ−= + + + + ≈ + ∫� , with

0 ,

1( , ) : ( , ) , k k

z k n nn nz z

nψ ϕ η= = . I call this a virtual integral [1] due to the fact that the integrand

normally cannot be precisely stated although its value can be easily computed. However,

when ( )z t can be derived in closed form, 0( , ) ( ( ), )z t z t tψ ϕ= and

( )1 1

0 0

( ) ( , ) ( ), (1) (0)z z t dt z t t dt z zλ ψ ϕ= = = −∫ ∫

Example 1 : 2( ) zz z e t it= + = +F ,

21( )

2i zω ω ω= − + =G .

2( , )

1z z

t iz t

eϕ

+=

+,

2

( ) (1 )zt i

i eωϕ

ω

+=

− + ⋅ +.

1

0

( ) ( ( )) ( ) .1373 .2264I z dz z t z t dt iγ

ω ω ′= = ⋅ ≈ − +∫ ∫

The ( )zF contour is in green. Integral value is the red vector. And the blue contour is the

following: 0

( ) ( ( )) ( )

t

C t z z dω τ τ τ′= ∫ , :0 1t → .

Example 2 : ( ) 2 2, ( ) 0 , ( ) 4 2 1z t z t t t itζ ζ= − = = + +F ,

( ) ( 2 ) (2 1) 0 , ( )u v i u z x iy u ivω ω= + + + − = + = +G . 0 0

1 , .5 .75z iω= = − + .

1

0

( ) ( ( )) ( ) .7803 1.0858I z dz z t z t dt iγ

ω ω ′= = ⋅ ≈ − +∫ ∫

Example 3 : 2 2( , ) (2 1) 0z t z t it= − − + =F , ( ) 0e i zωω ω ω= + − =G .

( ) .9713 1.5080z dz iγ

ω ≈ − −∫

Example 4 : 2:F i zω ω+ = and

2( , ) 0 , .6742 .0725zz t ze c t c i= + − = ≈ −F , giving

0 0.5 , .5 .5z i iω= − + = + ⇒ .3612W f dz

γ

= ≈ −∫ � . The red VF is the force field F , the green

VF is the field of contours.

The following are simple topographical images of ( )zλ where ( )z t is unavailable or tedious to

unscramble.

Example 5 : 3 2( , ) ( ) 0z t iz t it cΦ = − + + = ,

2

2( , )

3

t iz t

izϕ

+= .

1

0

( ) ( , )z z t dtλ ψ= ∫ :

(The z in the integral is actually 0z )

-1.2<Re(z)<1.2 , -1.2<Im(z)<1.2

Example 6 :2( , ) ( ) ( ) 0z t Sin zt iz t it cΦ = − − + + = ,

2 ( )( , )

( )

t i zCos ztz t

tCos zt iϕ

+ −=

−.

1

0

( ) ( , )z z t dtλ ψ= ∫ :

-15<Re(z)<15 , -15<Im(z)<15

Example 7 : 2 2( , ) ( ) 0zz t zte z t it cΦ = − − + + = ,

2( , )

(1 ) 2

z

z

t i zez t

te z zϕ

+ −=

+ −.

1

0

( ) ( , )z z t dtλ ψ= ∫ :

-15<Re(z)<10, -15<Im(z)<15

Example 8 : 3 2( , ) ( ) 0z t z t iz t it cΦ = − − + + = ,

3

2

2( , )

3

t i zz t

z t iϕ

+ −=

−.

1

0

( ) ( , )z z t dtλ ψ= ∫ :

-5<Re(z)<5, -5<Im(z)<5

Example 9 : 2 2( , ) ( ( 1)) ( ) 0z t Ln z t z t it cΦ = + − − + + = ,

2

2 1 /( 1)( , )

1 2

t i tz t z

zϕ

+ − += ⋅

−.

1

0

( ) ( , )z z t dtλ ψ= ∫ : -2<Re(z)<2, -2<Im(z)<2

Example 10 : ( ) ( ) ( ) 2, ; ( ) ( ) ( ) 0x y t x Sin y i y Cos x t it cΦ = + + − − + + = ,

2 ( ) 1 2 ( )( , )

1 ( ) ( ) 1 ( ) ( )

t Cos y tSin xz t i

Sin x Cos y Sin x Cos yϕ

− −= + ⋅

− −.

1

0

( ) ( , )z z t dtλ ψ= ∫ :

-10<Re(z)<10, -10<Im(z)<10

Example 11 : ( ) 2, ; ( ) ( ) ( ) 0x y t xCos y iySin x t it cΦ = + − + + = ,

1

0

( ) ( , )z z t dtλ ψ= ∫ :

10<Re(z)<10, -10<Im(z)<10

Compositions of Sets of Implicit Contours

Suppose the set of Zeno contours arising from an implicit function is designated:

[ ] ( ): , 0 ( , )m m m

dzz t z t

dtγ ϕΦ = ⇒ =

Then compose sets of contours in the following manner:

[ ] [ ] ( ) *

1 2 1 2: ( , ), ( , )dz

z t t z tdt

γ γ ϕ ϕ ϕ= =�

This leads to virtual integrals

1

* *

0

( ) ( , )z z t dtλ ψ= ∫

Example 12 : [ ] ( ) 3 2

1 1: , ( ) 0z t z t itγ ζΦ = − + + = , [ ] ( ) 2

2 2: , ( ) 0zz t te z t itγ ζΦ = + − + + =

1 22

2 2 ,

3 1

z

z

t i t i e

z teϕ ϕ

+ + −⇒ = =

+.

1

* *

0

( ) ( , )z z t dtλ ψ= ∫ :

-.5<Re(z)<2 , -1<Im(z)<2.5

Example 13 : [ ] ( ) 2 2

1 1: , ( ) 0z t tz z t it cγ Φ = + − + + = , [ ] ( ) 2 2

2 2: , ( ) 0ztz t e iz t it cγ Φ = − − + + =

2

1 2

2 2 ( , ) , ( , )

2 1 2

tz

tz

t i z t i zez t z t

tz te ziϕ ϕ

+ − + −⇒ = =

+ −.

1

* *

0

( ) ( , )z z t dtλ ψ= ∫ :

-3<Re(z)<8 , -10<Im(z)<10

Integrals of Contours

Suppose for an implicit function ( ), 0z tΦ = involving an undetermined constant c we consider

the set of contours { ( )z t } . Individual contours are determined by 0

( )c zζ ζ= = so that we

may graph

1

0

( ) ( , )z t dtν ζ ζ= ∫ topographically over a part of the ζ -plane (which lies

superimposed over the z-plane).

Example 14 : 3 2( , ) ( ) 0z t iz t it cΦ = − + + = (example 5).

1

0

( ) ( , )z t dtν ζ ζ= ∫ :

-1.2<Re(z)<1.2 , -1.2<Im(z)<1.2

Example 15 : 3 2( , ) ( ) 0z t z t iz t it cΦ = − − + + = (example 8).

1

0

( ) ( , )z t dtν ζ ζ= ∫ :

-5<Re(z)<5, -5<Im(z)<5

Implicit Contour Integrals

We wish to evaluate the integral ( )f z dzγ∫ if we are given ( ) ( )f z f x iy= + but the contour γ

lies implicit (and unrevealed) in ( ), 0z tΦ = :

Example 16 : Given ( ) ( ) sin( )f x iy xCos y iy x+ = + and the contour derived from

2( ) 0zte z t it ζΦ = + − + + = , 0

(0)z z= , evaluate I = ( )f z dzγ∫ .

The last condition exists when 0zζ = . We have

( ) ( )(1) 1

0

( ) ( ) ( ) ( ) ( ), ( )

z

f x iy dz f x t iy t x t y t dtζ

ϑ ζ ϕ= + = +∫ ∫ where 2

1

z

z

t i e

teϕ

+ −=

+. A simple

topographical image results from graphing ( )ϑ ζ over part of the complex plane.

-10<Re(z)<10, -10<Im(z)<10

[1] J. Gill, Informal Notes: Zeno Contours, Parametric Forms, & Integrals , Comm. Anal. Th. Cont. Frac., Vol XX

(2014)