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Reduced Exponential

  1. Aug 4, 2007 #1
    "Reduced Exponential"

    I am interested in what I call the "reduced exponential"
    Sum_i=1 to infinity x^(i-1) / i!
    where x is a general element in an algebra of interest.

    Only when x is invertible is the reduced exponential equivalent to (exp(x)-1) /x .

    Obviously we have a "reduced log", the inverse of the reduced exponential.

    Does anybody of any work or formulae involving this construct? TIA.
     
  2. jcsd
  3. Aug 4, 2007 #2

    CompuChip

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    How do you mean: "if x is invertible"?
    If x is a number the series is always equal to (exp(x) - 1)/x, unless x= 0 in which case it converges to zero. If not, the notation with the division doesn't make sense.

    Where did you encounter this function?
     
  4. Aug 4, 2007 #3
    "How do you mean: "if x is invertible"?"

    x is an element of a general algebra, not merely a real or complex number but a multivector or matrix or similar such object that can be raised to integer powers and summed and so exponentiated. I've encountered this in quantum mechanics ..
     
  5. Aug 5, 2007 #4

    CompuChip

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    OK, it is possible to define the exponential in such cases, but then I would write
    [tex](\exp(x) - 1) x^{-1}[/tex] (or [tex]x^{-1} (\exp(x) - 1)[/tex], though I think there is no difference here) instead of the division.
     
  6. Aug 5, 2007 #5

    D H

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    Writing [tex]x^{-1}[/tex] instead of dividing by [tex]x[/tex] doesn't help. What if x is not invertible? For example, the matrix
    [tex]x = \bmatrix 0 & 1 \\ 0 & 0 \endbmatrix[/tex]
    has no inverse but certainly has a "reduced exponential" as defined in the OP: [tex]\sum_{i=1}^{\infty} \frac {x^{i-1}}{i!} = \bmatrix 1 & 1/2 \\ 0 & 1 \endbmatrix[/tex]
     
    Last edited: Aug 5, 2007
  7. Aug 5, 2007 #6

    CompuChip

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    Doesn't help for what? I was just pointing out a notational inconvenience in
    which still holds, as in the example you gave the sum evaluates to the identity which is not even close to exp(x) - 1 = x.

    The question was
    which I must admit, I can't recall having seen or used anywhere.
     
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