andydude Wrote:Yes, the main difference between the one above and Woon's original expansion is that his expansion uses two (-1) factors, whereas I use one (-1) with two exponents... basically the same.
For his original paper, you can download it from arxiv.org:
http://arxiv.org/abs/hep-th/9707206 (click on PS or PDF)
and his formulas are at #18 (for the D operator), #71 (for any operator).
Yes I already had a look at it.
His derivation was meant for operators not for functions.
His paper mainly deals with fractional differentiation, there you have the differentiation operator D. An operator is usually a linear map in a function space, in this case D maps a function onto its derivative and it is linear as it has the property \( D(f+g)=D(f)+D(g) \) and \( D(\alpha f)=\alpha D(f) \). So the derivation of Woon is not directly applicable to functions instead of operators.
Of course one can chose the power derivation matrix as operator. However the coefficients of the continuous iteration of power series with fixed point at 0 can be obtained in a finite manner. I.e. with no limits involved. Which is not the case in Woons expansion. There it merely works if \( A \) is upper triangular with a diagonal of 1's and \( w=1 \). Because in this (parabolic) case a truncated \( A-I \) is nilpotent \( (A-I)^n=0 \) and so the involved sum is finite.
But already it can not be used already for the hyperbolic case.
Where you can use the diagonalization instead to obtain a finite solution.
Also with infinite sums there is always the question of convergence, which can sometimes not be established for matrices though for example \( (1+x)^t \) can be defined also for real \( x>1 \) despite the series is converging for only \( |x|<1 \).