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Chapter1.Appendix.Dirac.Delta

Chapter1.Appendix.Dirac.Delta - IV-30 The Dirac Delta...

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IV-30 Dirac Delta Function In one dimension, į (x-x o ) is defined to be such that: + * 0 i f x o is not in [a,b]. ± a to b f(x) į (x-x o )dx ± * ½f(x o ) if x o = a or b; * f(x o ) if x o İ (a,b). . The Dirac Delta Function, į (x-x o ) Properties of į (x-x o ): (you should know those marked with * ) * 1. į (x-x o ) = 0 if x ± x o * 2. ± - ± to + ± į (x)dx = 1 3. į (ax) = į (x)/|a| * 4. į (-x) = į (x) 5. į (x²-a²) = [ į (x-a) + į (x+a)]/(2a); a ² 0 6. ± - ± to + ± į (x-a) į (x-b)dx = į (a-b) + ------------------------------------------------------------------------------------------------------------------------------------------ - * * 7. į (g(x)) = ³ i į (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) į (x-a) = f(a) į (x-a) 9. į (x) is a "symbolic" function which provides convenient notation for many mathematical expressions. Often one "uses" į (x) in expressions which are not integrated over. However, it is understood that eventually these expressions will be integrated over so that the definition of į (box above) applies. 10. No ordinary function having exactly the properties of į (x) exists. However, one can approximate į (x) by the limit of a sequence of (non-unique) functions, į n (x).
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Chapter1.Appendix.Dirac.Delta - IV-30 The Dirac Delta...

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