Generalization of The Abel Transform To Discontinuous F(y)
Consider the case where is discontinuous at, where it abruptly changes its value by a finite amount . That is, and are defined by . Such a situation is encountered in tethered polymers (Polymer brush) exhibiting a vertical phase separation, where stands for the polymer density profile and is related to the spatial distribution of terminal, non-tethered monomers of the polymers.
The Abel transform of a function f(r) is under these circumstances again given by:
Assuming f(r) drops to zero more quickly than 1/r, the inverse Abel transform is however given by
where is the Dirac delta function and the Heaviside step function. The extended version of the Abel transform for discontinuous F is proven upon applying the Abel transform to shifted, continuous, and it reduces to the classical Abel transform when . If has more than a single discontinuity, one has to introduce shifts for any of them to come up with a generalized version of the inverse Abel transform which contains n additional terms, each of them corresponding to one of the n discontinuities.
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