A Note on Integrals & Hybrid Contours in the Complex Plane

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A Note o Itegrals & Hybrid Cotours i the Complex Plae Joh Gill July 4 Abstract: Cotour itegrals ca be expressed graphically as simple vectors arisig from a secodary cotour.

1 A Note o Itegrals & Hybrid Cotours i the Complex Plae Joh Gill July 4 Abstract: Cotour itegrals ca be expressed graphically as simple vectors arisig from a secodary cotour. We start with a well-behaved (cotiuous, aalytic, etc.) fuctio i the complex plae: () ϕ( z, t ), with z S a covex set i C ad t [,]. ϕ( z, t) S z S ad t [,] Let k () g ( z) = z + η ϕ( z, ), with g, ( z) S, < k k, k, Require < η, < η, < < η, = ad limηk, =, where k =,,...,. G ( z) g ( z) Set,, k =, Gk, ( z) gk, ( Gk, ( z) ) = ad G ( z) = G, ( z) with G( z) = lim G ( z), whe that limit exists. Writte i a alterate iterative form k (3) z = z + η ϕ( z, ), z α, k, k, k, k,, the distributio of poits forms a Zeo cotour (4) { } γ ( α) = z γ ( α) = lim γ ( α). k, k =

2 The word Zeo deotes the ifiite umber of actios required i a fiite time period if η k, describes a partitio of the time iterval [,]. ϕ( z, t ) is associated with a uique time-depedet vector field (TDVF): (5) F : F( z, t ), where F( z, t) = ϕ( z, t) + z. Uder beig coditios (3) admits a equivalet closed form: dz (6) γ ( α) : z = z( t), = ϕ( z, t), z() = α. dt Now suppose aother well-behaved fuctio f( z, t) S is itroduced. Set (7) (8) ϕ * ( z, t) = f( z, t) ϕ( z, t) ad create a ew, hybrid cotour i the followig way: z = z + η ϕ ( z, ), z α, * * * k k, k, k, k,, * * γ α = ad γ ( α) γ ( α) as with * { * ( ) zk, } * Observe that ϕ is a fuctio of poits o the origial Zeo cotour. So that we ow have two cotours: γ( α ) (i gree) ad γ * ( α ) (i red) that are siamese, i.e., origiatig at the same poit. The uderlyig TDVF is illustrated by vector clusters (black for t=, gree for t=):

3 Example : k k k Rewrite (3) as f( zk,, ) ( zk, zk, ) = f( zk,, ) ϕ( zk,, ) k= k= (9) * k * fdz = α + lim ϕ ( zk,, ) α G ( α) α = k= Thus the value of the itegral is essetially the vector coectig α to G * ( α ).

x ( y t ) I Example above ϕ = + i + t + t ϕdz.667 + 4.6i, f = ϕ ad α = +.5i.

4 x ( y t ) I Example above ϕ = + i + t + t ϕdz i, f = ϕ ad α = +.5i. The If f = ϕ, a very well-behaved fuctio, () γ dz ( ) = ( ) = ( ( )) or ϕ z dz ϕ z dt ϕ z t dt dt ( z) z() ϕ( z) dz for aalytic ϕ. z The otatio fdz represets f ( z ) dz or f( z( t), t) ϕ( z( t), t) dt

63i Example 3: A simple case where direct evaluatio

5 Example : ϕ = xcos( yt) + iysi( xt), f = ( x + t) + i( x y t), α = i fdz i Example 3: A simple case where direct evaluatio is easily doe: ϕ ( z) = z z( t) t = αe, (, ) f z t = zt, α =.5( + i) fdz.995i

6 Example 4: ϕ = xcos( xt + y) + ysi( xt y) i, f = Si( x + yt) Cos( x yt) i, α = 5( + i) fdz i

7 Virtual Itegral vs secodary cotour: Assume ϕ = ϕ( z). Previously, a virtual itegral was established i the followig way: γ ( ) α : z, = α + ϕ( α) + ϕ( z, ) + ϕ( z, ) + + ϕ( z, ) Now, by slight of had, defie k mk, m m ( ) [ ] ( ) ( ) ψ α, t, t, ad ψ α, lim ϕ z, with ψ α, t dt defied: 3 ( α) α = ψ α, + ψ α, + ψ α, + + ψ α, ψ ( α, ) G t dt So that ψ ( α, t) dt = G ( α) α. Uder perfect circumstaces γ( α ): z = z( t) has a pleasat closed form ad ψ ( α, t) = ϕ( z( t)), α = z (). For example, ϕ( z, t) = zt ϕ( z( t)) = αte = ψ ( α, t ) t Therefore λ( α) = ϕ( z( t)) dt = G ( α) α, whereas, i this ote, * ϕ α α ( z( t)) dt = G ( ). However, it is usually the case that ψ ( α, t ) caot be easily described as ϕ( z( t )) ad λ( α) = ψ ( α, t) dt = G ( α) α has a virtual itegrad. The otatio ϕ * α α ( z( t)) dt = G ( ) is used eve whe z( t ) is idescribable.

Example 5: ϕ = xcos( yt) + iysi( xt), α =.5 i λ( α) = ψ ( z, t) dt = G( α) α.789.764i, ad ϕ * α α ( z( t)) dt = G ( ).883 +.495i.

8 Example 5: ϕ = xcos( yt) + iysi( xt), α =.5 i λ( α) = ψ ( z, t) dt = G( α) α i, ad ϕ * α α ( z( t)) dt = G ( ) i. ( Results hold for ϕ( z, t )) Simplified, heuristic descriptio of the process: Whe the fuctios are aalytic ad itegratios i closed forms are possible, we have dz dζ = ϕ( z) z = z( t) ad = f( z) ϕ( z), leadig to dt dt dζ = f( z( t)) ϕ( z( t)) dt = f( z) dz ζ() ζ() = f( z) dz = f( z( t)) ϕ( z( t)) dt Results are valid for f = f( zt, ) ad ϕ = ϕ( zt, ).

9 Equatio solvig... Let us restrict our discussio to f = ϕ. The what has bee described previously fits the format T( ϕ, α) = β ϕ( z) dz = β That is to say T: C( S) S C where C( S ) is the set of complex fuctios cotiuous o the set S ad C is the complex plae. Example 6: ϕ( z) = ϕ( x + iy) = xcos( xy) + iysi( xy), with associated vector field F : F( z) = ϕ( z) + z. Thus Τ ( xcos( xy) + iysi( xy),.5 +.8) i = ϕ( z) dz i γ(.5+.8 i)

10 Example 7: Determie α satisfyig the equatio ( z α ) z( t) e z() e t = α = α. Therefore α e dz Τ, = i. From z dt = zdz = i α ±.366( + i) α, oe fids Example 8: Give z( t) α( t) t i, α.i = + + =, fid β. dz From ϕ ( z, t) = we see that dt dz z 4 t( + t) = α + 4 ti = + i = ϕ( z, t) dt + t + t. Hece 3+ t y + ( t + t) F : f( z) = x + i y + + t + t, a TDVF Ad ϕ ( α ) ( z( t)) dt = 4 + ti dt i

11 Example 9: z( t) ϕ = ad β = + i. Solve forα : Τ ( ϕ, α) = β. From z = t + α ad the problem looks like this: gives a solutio: α.68.68i. + α dz = + i. Thus z α dz =, dt z + α = α e + i Attemptig to solve Τ ( ϕ, α) = β for ϕ requires more effort ad may etail restrictig the fuctioal form.

12 Example : Solve for C : Τ ( Cz, i) = + i From dz Cz dt = oe obtais ( ) Ct z t = ie. Therefore C zdz = + i. Solvig for C requires umerical techiques for itrisic fuctios. C C ie i

13 Example : Solve for C: Τ ( icz,) = + 3i C = 6 + 4i Example : Give z( t) = Cα( t + ) ti ad α =, solve T( ϕ,) = 4 i for ϕ. ϕ = ( Cαt i) ad umerical computatios give C i

14 Fial Commets... Settig λ( α) = ψ( α, t) dt ad ρ( α) = f( G( α)), it is possible to write ( f z zf z ) dz ( f ) λ( α) = ρ( α) ( ) '( ) α ( α) ρ( α) Give a primary ad a secodary cotour, k γ( α ): z = z + η ϕ( z, ) ad k, k, k, k, * * * k γ ( α ): z = z + η T( z, ) k, k, k, k, T( z) dz ϕ( z) * = α G ( ) α Eough of this trivia. I m gettig bored!

Informal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane.

Informal Notes: Zeno Contours, Parametric Forms, & Integrals. John Gill March August S for a convex set S in the complex plane. 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