Implementing Dual Unit from Dual Numbers

Question

I'm trying to implement the dual unit $\epsilon$ defined by $\epsilon^2 = 0$ into Mathematica but I'm having trouble defining $\epsilon^n = 0$ for $n\geq 2$. In this answer they simply use dualE /: Power[dualE, 2] := 0 which works well for $n=2$ but not for any other $n\geq 2$. I've tried

dualE /: Power[dualE, n] := If[n>=2, 0, dualE^n]

but then I just get $\epsilon^n = \epsilon^n$ for any $n$. I can also define the function

Nilpotent[dualE, n] := If[n>=2, 0, Power[dualE, n]]

which then gives Nilpotent[dualE, n] = 0 for $n\geq 2$, but if I try

dualE /: Power[dualE, n] := Nilpotent[dualE, n]

this still doesn't give the desired result for $\epsilon^{n\geq 2}$! As a last ditch effort I tried Table[dualE /: power[dualE, n] := 0, {n, 2, 100}]; which at least gives $\epsilon^n = 0$ for $2\leq n \leq 100$ but still evaluates to $\epsilon^n$ for any non-integer values of $n$.

There must be a way to define $\epsilon^{n\geq2}=0$ and $\epsilon^{n<2}=\epsilon^n$ properly so any help would be much appreciated!

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score 3 · Accepted Answer · answered Jan 25 '19 at 02:02

3

You have to use a pattern _ for the second argument; moreover, it might also be a good idea to ensure that n is an integer.

dualE /: Power[dualE, n_Integer /; n >= 2] := 0

Now things seem to work:

(1 + dualE)^2 // Expand

1 + 2 dualE

answered Jan 25 '19 at 02:02

Henrik Schumacher

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Thanks, that's great for integer values of n. Can I make it so it works for non-integer values of n too? E.g. dualE^(5/2) = 0. – Thomas Jan 28 '19 at 01:31
1

I've worked it out now: dualE /: Power[dualE, n_] := 0 /; n>=2 does the trick! – Thomas Jan 28 '19 at 03:09

Anixx · Answer 2 · 2022-10-26T12:13:36.717

This way:

$Post = MatrixFunction[
      Function[ε, #], {{0, 0}, {1, 0}}] /. {{a_, 
        b_}, {0, a_}} -> 
      a + ε b /. {{a_, 0}, {b_, a_}} -> 
     a + ε b &;

UPDATE

The following variants will also work (and better):

$Post=Normal[Series[#, {ε, 0, 1}]]&;

or

$Post=(#/.ε->0)+ε(D[#, ε]/.ε->0)&;

(the last method is the most universal)

Use this way:

 In:=I^ε
Out:= 1 + (I ε Pi)/2
In:=Log[a + ε]
Out:=ε/a + Log[a]
In:=f[a + b ε]
Out:=f[a] + b ε f'[a]

Implementing Dual Unit from Dual Numbers

2 Answers2