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Abel Transform

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The Abel transform is an integral transform relating two functions f(x) and g(t). For 0<alpha<1, one convention is

f(x)=int_0^x(g(t)dt)/((x-t)^alpha)
(1)
g(t)=(sin(pialpha))/pid/(dt)int_0^t(f(x)dx)/((t-x)^(1-alpha))
(2)
=(sin(pialpha))/pi[int_0^t(df)/(dx)(dx)/((t-x)^(1-alpha))+(f(0))/(t^(1-alpha))].
(3)

The Abel transform is used in calculating the radial mass distribution of galaxies (Binney and Tremaine 1987, p. 651; Arfken and Weber 2005, p. 1014) and inverting planetary radio occultation data to obtain atmospheric information as a function of height.

Bracewell (1999, p. 262) defines a slightly different form of the Abel transform given by

g(x)=A[f(r)]=2int_x^infty(f(r)rdr)/(sqrt(r^2-x^2)).
(4)

The following table gives a number of common Abel transform pairs (Bracewell 1999, p. 264), together with the exponential pair (Retana-Montenegro et al. 2012, p. 7). Here,

Pi_a(x)={1 for 0<x<a; 0 otherwise,
(5)

where Pi(x) is the rectangle function, and

M(x)=2pi[x^(-3)int_0^xJ_0(x)dx-x^(-2)J_0(x)]
(6)
=(pi^2)/(x^2)[J_1(x)H_0(x)-J_0(x)H_1(x)],
(7)

where J_n(x) is a Bessel function of the first kind and H_n(x) is a Struve function.

f(r)g(x)conditions
Pi_a(r)2sqrt(a^2-x^2)0<x<a
(a^2-r^2)^(-1/2)Pi_a(r)pi0<x<a
sqrt(a^2-r^2)Pi_a(r)1/2pi(a^2-x^2)0<x<a
(a^2-r^2)Pi_a(r)4/3(a^2-x^2)^(3/2)0<x<a
(a^2-r^2)^(3/2)Pi_a(r)3/8pi(a^2-x^2)^20<x<a
(a-r)Pi_a(r)asqrt(a^2-x^2)-x^2cosh^(-1)(a/x)
1/picosh^(-1)(a/r)a-x
delta(r-a)(2a)/(sqrt(a^2-x^2))Pi_a(x)
e^(-r/a)2xK_1(x/a)a>0, x>0
e^(-r^2/sigma^2)sigmasqrt(pi)e^(-x^2/sigma^2)sigma>0
r^2e^(-r^2/sigma^2)sigma(x^2+1/2sigma^2)sqrt(pi)e^(-x^2/sigma^2)sigma>0
(e^(-r^2/sigma^2))/(sigmasqrt(pi))(r^2-1/2sigma^2)x^2e^(-x^2/sigma^2)sigma>0
1/(b^2+r^2)pi/(sqrt(b^2+x^2))b^2+x^2>0
J_0(omegar)(2cos(omegax))/omegaomega>0
M(r)(8pi^4)/(omega^3x^2)sin^2((xomega)/(2pi))omega>0

Here, K_1(x) is the modified Bessel function of the second kind, and the exponential pair has the limiting value g(0)=2a. Fitted approximations in terms of the incomplete gamma function for the transforms of f(r)=e^(-(r/a)^p) with a>0 and 0.05<=p<=5 are given by Dhar (2021a, pp. 4586-4587; 2021b, p. 5468).

See also

Fourier Transform, Hilbert Transform, Integral Equation

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References

Abel, N. H. Oeuvres Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint Corp., pp. 11 and 97, 1988.Arfken, G. and Weber, H. J. Mathematical Methods for Physicists, 6th ed. Orlando, FL: Academic Press, p. 1014, 2005.Binney, J. and Tremaine, S. Galactic Dynamics. Princeton, NJ: Princeton University Press, p. 651, 1987.Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 262-266, 1999.Dhar, B. K. "High-Accuracy Analytical Solutions for the Projected Mass (Counts) and Surface Density (Brightness) of Einasto Profiles." Mon. Not. R. Astron. Soc. 504, 4583-4588, 2021a. https://doi.org/10.1093/mnras/stab1029.Dhar, B. K. "Erratum: High-Accuracy Analytical Solutions for the Projected Mass (Counts) and Surface Density (Brightness) of Einasto Profiles." Mon. Not. R. Astron. Soc. 505, 5468, 2021b. https://doi.org/10.1093/mnras/stab1760.Hilfer, R. (Ed.). Applications of Fractional Calculus in Physics. Singapore: World Scientific, pp. 3-4, 2000.Liouville, J. "Memoire sur quelques quéstions de géométrie et de mécanique, et sur un nouveau genre pour réspondre ces quéstions." J. École Polytech. 13, 1-69, 1832.Lützen, J. Joseph Liouville, 1809-1882. Master of Pure and Applied Mathematics. New York: Springer-Verlag, p. 314, 1990.Retana-Montenegro, E.; Van Hese, E.; Gentile, G.; Baes, M.; and Frutos-Alfaro, F. "Analytical Properties of Einasto Dark Matter Haloes." Astron. Astrophys. 540, A70, 2012. https://doi.org/10.1051/0004-6361/201118543.Whittaker, E. T. and Robinson, G. The Calculus of Observations: A Treatise on Numerical Mathematics, 4th ed. New York: Dover, pp. 376-377, 1967.

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Abel Transform

Cite this as:

Weisstein, Eric W. "Abel Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelTransform.html

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