Formal delta

The formal delta over a field is the Laurent series¹ #m/def/fcalc

given by the Fourier series expansion of the Dirac delta.

Properties

Let be a vector space over . Let and . Finally let and . Then in

Let such that exists and . Finally let , , and . Then in

Note these fail for non-integer powers.

Proof of 1–6

First we prove ^P1. Consider the special case . Then

whence follows ^P1 by linearity. The proof of ^PA is similar. Let . Then

Then ^P3 and ^PC follow by taking appropriate derivatives, and ^P2 and ^PB are special cases.

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