Class_2 - PHYS809 Class 2 Notes The Dirac delta function δ...

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Unformatted text preview: PHYS809 Class 2 Notes The Dirac delta function ( ) δ r This has the properties ( ) 0 if , δ- = ≠ r a r a ( ) 1, allspace dV δ = ∫ r ( ) ( ) ( ) if 0 otherwise. V f V f dV δ ∈- = ∫ a a r r a This is useful for denoting the charge distribution due to point charges. The Dirac delta function can be represented in a number of ways. It is often convenient to think of the delta function in terms of a limit of a one parameter set of functions, with the integral over all space of a function equal to unity. For example, in 1-dimension, ( ) 2 2 1 2 1 1 lim . 2 x x e α α δ α π- → = We can use this form to show that ( ) ( ) ( ) , i i i x x f x f x δ δ- = ′ ∑ where x i is a zero of f ( x ). Now ( ) ( ) 2 2 1 2 1 1 lim . 2 f x f x e α α δ α π- → = Near a zero of f ( x ), we have ( ) ( ) ( ) . i i f x x x f x ′ =- + ⋯ Hence ( ) ( ) ( ) 2 2 2 1 2 1 1 lim . 2 i i f x x x i f x e α α δ α π ′-- → = ∑ Let ( ) , i i f x α β = ′ so that ( ) ( ) ( ) ( ) ( ) 2 2 1 2 1 1 lim . 2 i i i x x i i i i i i x x f x e f x f x β β δ δ β π-- →- = = ′ ′ ∑ ∑ Another useful representation of the delta function is ( ) 1 ....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.

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Class_2 - PHYS809 Class 2 Notes The Dirac delta function δ...

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