Abel Transform - Generalization of The Abel Transform To Discontinuous F(y)

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

f(r)=\left \Delta F-\frac{1}{\pi}\int_r^\infty\frac{d F}{dy}\frac{dy}{\sqrt{y^2-r^2}}.

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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