Informal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane.
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1 Iformal Notes: Zeo Cotours Parametric Forms & Itegrals Joh Gill March August 3 Abstract: Elemetary classroom otes o Zeo cotours streamlies pathlies ad itegrals Defiitio: Zeo cotour[] Let gk ( z = z + ηk ϕ( z where z S ad gk ( z S for a covex set S i the complex plae Require limηk = where (usually k = Set G ( z g ( z = Gk ( z gk ( Gk ( z = ad G ( z = G ( z with G( z = lim G ( z whe that limit exists The Zeo cotour is a graph of this iteratio The word Zeo deotes the ifiite umber of actios required i a fiite time period if ηk describes a partitio of the time iterval [] Normally ϕ ( z = f ( z z for a vector field F = f The alterative otatio G z =L g ( z is also available Zeo cotours are a atural ad mior extesio of Euler s ( k Method for differetial equatios l Begi with η k = ad gk ( z z + ϕ( z z S gk ( z S The we have with ϕ( z cotiuous o a domai S ad G ( z = z + ϕ( z + ϕ( G ( z + ϕ( G ( z + + ϕ( G ( z Now imagie a fuctio k ψ ( z t t [ ] ad ψ z lim ϕ ( Gmk m( z with ψ ( defied m z t : 3 G( z z = ψ z + ψ z + ψ z + + ψ z ψ ( z t Ad for t irratioal ψ ( z t = lim ψ ( z t r for ratioal t r The existece of this fuctio (ad the itegral is tr t equivalet to the covergece of the Zeo cotour ψ ( form ca be murky at times but very simple uder the right circumstaces z t is more a virtual itegral sice its aalytical ~ ~
2 (i Zeo Cotour to Parametric Form: Write the recurrece sequece as k k k z ( z ( ( z ( z( z ϕ = + = Assumig z = z( t oe cocludes z dz = ϕ = k t k ( z( ϕ ( z( t t [] dz Whe coditios allow: ψ( z t = ϕ( z( t = ad ψ ( z t = z( z( Example: ϕ ( z = z produces ( dz z = which gives z z( t = ad that i tur gives zt ψ z t = λ( z = z ( z Here ψ ( z t = z = z ( zt z z z z Example: ϕ ( z = e Here z( t = z L( e t ad ψ ( z t = e ( e t z α z( t + α Example: ϕ( z = However z( t z = t + ( α β L allows o closed z β z + α formulatio of ψ ( z t although z( + α ψ( z t = + ( α β L z + α dx dy Example: ϕ ( z = ( x + y + i( x y We have = x + y = x y which ca be solved x( t = Ce + C e ad t t y( t = Ce + C e with Ck = C ( k x y t t 3 4 ~ ~
3 dx dy Example: ϕ ( z = xcos( y + iysi( x givig = xcos( y = ysi( x a system ot solvable i closed form if at all The two cotours use η = η = the latter termiatig at oe of may attractors o the real axis (ii Parametric Form to Zeo Cotour: Give z = z( t t describe ϕ ( z or ϕ ( z t We must have z( = z dz Example: z( t = ze αt = αz = ϕ( z αt dz α Example: z( t = ze + βt = αz t e + β = αz + β( αt = ϕ( z t Therefore the k geeratig fuctios are gk ( z = z + ( z ( α + β α ad the cotour is a pathlie rather tha a streamlie give by G( z lim ( g g g ( z = The uderlyig force field is ( + α z + β( αt = ϕ( z t + z = f( z t f( z = ( + α z + β( α ~ 3 ~
4 Example: z( t z ( = + t dz z = = ϕ( z t givig a pathlie where the vector fields + t 3+ t f( z t = z f( z = z + t g ( z = z + G ( z 4z k + k ll A little visual backgroud: Suppose f ( z α ρ z α ρ By drawig a pair of cocetric circles < < i a covex regio S = { ζ ζ α = z α } ad c ( z = { ζ : ζ α = ρ z α } c ( z : C it is ot difficult to establish the followig: g( z α z α f ( z z leadig to ( C g z ( α ρ z α for large C G z α ρ z α Thus ( ( Writig C( ρ C ( ρ C ( ρ = e C ( ρ C( ρ If C = C = the the last expressio approximates e ( ρ ad covergece occurs Thus η k = forces the cotour to termiate at a attractor For η k = whe the differetial equatios described previously ca be solved early termiatio follows easily Zeo cotours follow streamlies of course ( ( ( ( ( Writig g z = z + ( f z z ad the g z α = z α + ( f z α for f( z α < ρ z α a simple vector aalysis shows that i a approximate sese ~ 4 ~
5 + ρ ρ ( z α G z α z α e e For large values of Thus the leadig tip of the cotour is trapped i a aulus that is quite small if z is close to α Example: = + z = 7 + 3i = η = vs f( z z z η = ( Z = Z z t = z t z = Z( z z III Cosider the fuctio defied by = λ( z ψ( z t The image i the z-plae (red is γ ( s ad the image i the w-plae (gree is λ( γ ( s ~ 5 ~
6 Example: ϕ ( z = Cos( z z provides the followig mappigs of circles (scales are oe uit: z( z( e Aother example is ϕ ( z = z z with λ( z = Ψ ( z t = z( e + e A o-aalytic example: ϕ( z = ϕ( x + iy = xcos( y + ixsi( y ~ 6 ~
7 Ad aother: ϕ( z = ϕ( x + iy = xcos( x + y + iysi( y x Also ϕ( z ϕ( x iy ycos( x ixsi( y = + = + for ( z = 3 V The Zeo itegral is loosely associated with Ergodic theory i the followig way: ψ( z t G ( z z = ϕ ( Gk ( z a average value of { ( } ( ϕ Gk z sequece { Gk z } ( are poits distributed alog the emergig Zeo cotour where the ~ 7 ~
8 Example: Set ϕ ( z = z z for z = + i ad = : { } The (approximate Zeo cotour is i purple while the distributio ϕ ( Gk z ( is i gree The poit λ( z lies i quadrat III withi the arc of the distributio: λ ( + i 87 + i( 3 Aother itegral worthy of metio ad easily obtaied [] is γ ( ϕ ( z ϕ ( z dz = lim ϕ ( Gk ( z so that the itegral where the itegrad is basic for geeratig the Zeo cotour upo which it { ( } is evaluated provides the average value of the sequece ϕ ( Gk z sequece { } G ( z k is geerated usig the fuctio ϕ ot ϕ Note that the ~ 8 ~
9 Example: ϕ ( x + iy = xsi( x + y + iycos( x y z = + i The γ ϕ + ϕ i ( ( Gk z Pathlies coicide with streamlies if the uderlyig force field remais costat Whe it does ot pathlies may or may ot lie close to streamlies I the laguage of Zeo cotours C C pathlies are computable i the followig format: gk ( z = z + ϕk ( z = z + ( fk ( z z dz G ( z = g g g ( z Agai whe ϕ is well-behaved = ϕ( z( t t may be solvable ad t elimiated from the equatios Ad if fk C f i the sese that fk ( z f( z < εk where σ = εk the whe f ( z α < ρ z α α k k k k C gk ( z α = z α + ( fk ( z f( z + f( z z C C C gk ( z α z α ρ z α εk + + k C µ = ( ρ Thus G z Ad C = provides covergece as before α ad ρk ρ < G ( z α as before: C g ( z α µ z α + ε k k ( ( ( C ρ C ρ C ρ C ( α µ z α + εk + σ e Example: ϕ = x y dz i + t + + t = y leadig to y = x x x ~ 9 ~
10 Example: ϕ ( z t = xcos( x + y t + iysi( x + y t With vector fields ( ( f( zt = xcos( x + y t + x + i ysi( x + y t + y f( z = x( + Cos( x + y + iy( + Si( x + y Referece: [] J Gill Zeo Cotours i the Complex Plae Comm Aal Th Cot Frac XIX ( wwwjohgillet ~ ~
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