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M402-iv.not.page 30.Fall.02

# M402-iv.not.page 30.Fall.02 - IV-30 The Dirac Delta...

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IV-30 Dirac Delta Function In one dimension, δ (x-x o ) is defined to be such that: + * 0 if x o is not in [a,b]. a to b f(x) δ (x-x o )dx * ½f(x o ) if x o = a or b; * f(x o ) if x o ε (a,b). . The Dirac Delta Function, δ (x-x o ) Properties of δ (x-x o ): (you should know those marked with * ) * 1. δ (x-x o ) = 0 if x x o * 2. - to + δ (x)dx = 1 3. δ (ax) = δ (x)/|a| * 4. δ (-x) = δ (x) 5. δ (x²-a²) = [ δ (x-a) + δ (x+a)]/(2a); a 0 6. - to + δ (x-a) δ (x-b)dx = δ (a-b) + ------------------------------------------------------------------------------------------------------------------------------------------ - * * 7. δ (g(x)) = i δ (x-x oi )/|dg/dx| x=xoi where g(x oi ) = 0 and dg/dx exists at and in a region around x oi . . ------------------------------------------------------------------------------------------------------------------------------------------ - * 8. f(x) δ (x-a) = f(a) δ (x-a) 9. δ (x) is a "symbolic" function which provides convenient notation for many mathematical expressions.
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