the cylindrical radius, then we may want to know the projection of that function onto a plane parallel to the z axis. Without loss of generality, we can take that plane to be the yz plane, so that which is just the Abel transform of f(ρ, z) in ρ and y. A particular type of axial symmetry is spherical symmetry. In this case, we have a function f(r), where r2 = x2 + y2 + z2. The projection onto, say, the yz plane will then be circularly symmetric and expressible as F(s), where s2 = y2 + z2. Carrying out the integration, we have which is again, the Abel transform of f(r) in r and s. Verification of the inverse Abel transform Assuming is continuously differentiable and , drop to zero faster than , we can set and . Integration by parts then yields Differentiating formally, Now substitute this into the inverse Abel transform formula: By Fubini's theorem, the last integral equals 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. Relationship to other integral transforms Relationship to the Fourier and Hankel transforms The Abel transform is one member of the FHA cycle of integral operators. For example, in two dimensions, if we define A as the Abel transform operator, F as the Fourier transform operator and H as the zeroth-order Hankel transform