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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## 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.

### Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online