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Frabjous
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There are many representations of the delta function. Is there a place/reference that lists AND proves them? I am interested in proofs that would satisfy a physicist not a mathematician.
I wonder what you would consider a proof of a representation . A delta function is a something with certain well described properties. The word distributions comes to mind (out of the dust of ages). Showing that the representation exhibits such properties is just that: a showing.caz said:representations of the delta function. Is there a place/reference that lists AND proves them
What are the properties you refer to?caz said:I am actually comfortable with the delta function. It’s just that I am many years past grad school and was reading something (Intermediate Quantum Mechanics by Bethe) that piqued my interest in how to to show that these representations have the correct properties.
A delta function, also known as the Dirac delta function, is a mathematical function that is defined as zero everywhere except at one point, where it is infinite. It is often used as a tool in mathematics and physics to model point-like objects or phenomena.
A delta function is typically represented as δ(x) or δ(x-a), where 'a' is the point at which the function is non-zero. It can also be represented graphically as a spike at the point where it is non-zero.
The integral of a delta function is equal to 1. This is because the delta function is defined as zero everywhere except at one point, where it is infinite. When integrated over its entire domain, the function "picks out" the value at that point, which is equal to 1.
The delta function has various physical interpretations, depending on the context in which it is used. In physics, it is often used to model point particles or point charges. In signal processing, it represents an instantaneous impulse. In probability theory, it represents a point probability distribution.
The delta function is a useful tool in mathematics and science because it allows us to simplify calculations and model real-world phenomena. It is commonly used in Fourier analysis, differential equations, and quantum mechanics, among other fields. It also has applications in engineering, economics, and other areas of study.