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

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1 PHYS809Class20Notes 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 ( ) ( ) ( ) ( ) ( ) 0 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 0 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 - + = - - (16.4) The solutions to the homogeneous equations are the modified Bessel functions ( ) m I k ρ and ( ) .
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  • Fall '11
  • MacDonald
  • Coordinate system, Polar coordinate system, Dirac delta function, Cylindrical coordinate system, cylindrical polar coordinates

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