OK, the Dirac delta function has the following properties:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1[/tex]

and

[tex]\int_{ - \infty }^{ + \infty } {f({x_1})\delta ({x_1} - {x_0})d{x_1}} = f({x_0})[/tex]

which is a convolution integral. Then if [tex]f({x_1}) = \delta (x - {x_1})[/tex]

we get

[tex]\int_{ - \infty }^{ + \infty } {\delta (x - {x_1})\delta ({x_1} - {x_0})d{x_1}} = \delta (x - {x_0})[/tex]

which is a self-convolution and is seen on the wikipedia site for the Dirac delta function. But this can be iterated to get:

[tex]\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {\delta (x - {x_2})\delta ({x_2} - {x_1})\delta ({x_1} - {x_0})d{x_2}d{x_1}} } = \int_{ - \infty }^{ + \infty } {\delta (x - {x_1})\delta ({x_1} - {x_0})d{x_1}} = \delta (x - {x_0})[/tex]

And if iterated an infinite number of times we get:

[tex]\int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } { \cdot \cdot \cdot \int_{ - \infty }^{ + \infty } {\delta (x - {x_n})\delta ({x_n} - {x_{n - 1}}) \cdot \cdot \cdot \delta ({x_1} - {x_0})d{x_n}d{x_{n - 1}} \cdot \cdot \cdot d{x_1}} } } = \delta (x - {x_0})[/tex]

which is seen in Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th Edition, by Hagen Kleinert, page 91, eq 2.17

So all the Dirac delta functions seem to form a space closed under self-convolution. And I wonder what other algebraic structures are implied by the above. And particularly I wonder if there is any structure here that proves that the Dirac deltas must be complex. Or what other structures are necessary to make the deltas complex?

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# Algebraic structure of Dirac delta functions

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