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Manuscript Title: A method and a program for the numerical evaluation of the Hilbert transform of a real function.
Authors: O.E. Taurian
Program title: FHT
Catalogue identifier: ABVD_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 20(1980)291
Programming language: Fortran.
Computer: IBM 370/155.
Operating system: OS/MVT.
RAM: 4K words
Word size: 32
Keywords: General purpose, Other numerical, Hilbert transform, General polynomial Approximations, Cubic splines.
Classification: 4.12.

Nature of problem:
This program calculates the Hilbert transform of a continuous function. The evaluation of the matrix elements of the Green's function in quantum mechanics involves the evaluation of the Hilbert transform of the matrix elements of the Hamiltonian operator. The same type of problem appears in many other fields of applied physics. The program assumes that a cubic spline interpolation has been performed over the integration region. As the Hilbert transform function is evaluated at a point of the complex plane, the program checks first if the particular point needs special treatment. Then the corresponding analytical expressions for each interval are evaluated.

The function must be real and continuous over the integration interval. For complex functions two separate evaluation are necessary. In the appendix, it is shown how this program may be used in more general situations, for example when finite discontinuities are present.

Running time:
The running time depends on the number of points included in the integration interval. For the test run this number was equal to 200. The evaluation of 60 different points of the complex plane took 11 s on a IBM 370/155.