Class_16new - 1 PHYS809 Class 20 Notes Green function in...

Unformatted text preview: 1 PHYS809 Class 20 Notes Green function in cylindrical polar coordinates In cylindrical polar coordinates ( ) , , , z ρ φ the Green function is a solution of ( ) ( ) ( ) ( ) 2 4 , , , , , . G z z z z π ρ φ ρ φ δ ρ ρ δ φ φ δ ρ ′ ′ ′ ′ ′ ′ ∇ = ---- (16.1) If φ and z are unrestricted, we can represent the last pair of Dirac delta functions by ( ) ( ) ( ) ( ) ( ) 1 1 1 , cos . 2 2 im ik z z m e z z e dk k z z dk φ φ δ φ φ δ π π π ∞ ∞ ∞ ′ ′-- =-∞-∞ ′ ′ ′ - =- = =- ∑ ∫ ∫ (16.2) We then represent the Green function in a similar way ( ) ( ) ( ) ( ) 2 1 , , , , , cos , , . 2 im m m G z z e k z z g k dk φ φ ρ φ ρ φ ρ ρ π ∞ ∞ ′- =-∞ ′ ′ ′ ′ ′ =- ∑ ∫ (16.3) Substitution into equation (16.1) gives that ( ) , , m g k ρ ρ ′ is a solution of ( ) 2 2 2 1 4 . m m g m k g π ρ δ ρ ρ ρ ρ ρ ρ ρ ∂ ∂ ′- + = -- ∂ ∂...
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