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Hilbert transforms in yukawan potential theory.

R J Duffin1

  • 1Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.

Proceedings of the National Academy of Sciences of the United States of America
|December 1, 1972
PubMed
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Researchers generalized the Hilbert transform for Yukawa potentials, extending harmonic function theory. This new transform, H(mu), uses Bessel functions and is a convolution type, offering new mathematical tools.

Area of Science:

  • Mathematical Physics
  • Harmonic Analysis
  • Integral Transforms

Background:

  • The classical Hilbert transform relates conjugate harmonic functions in the upper half-plane.
  • Harmonic functions satisfy Laplace's equation (Delta u = 0).

Purpose of the Study:

  • To generalize the Hilbert transform for equations beyond Laplace's.
  • Introduce a new transform, H(mu), applicable to Yukawa potentials.

Main Methods:

  • Replacing the Laplace equation with the Yukawa equation (Delta u = mu^2 u).
  • Generalizing the Cauchy-Riemann equations.
  • Analyzing the properties of the generalized transform, H(mu).

Main Results:

  • A generalized Hilbert transform, H(mu), was developed.

Related Experiment Videos

  • The kernel of H(mu) is expressed using the Bessel function K(0).
  • The new transform is confirmed to be of convolution type.

    • Conclusions:

    • The study successfully extends Hilbert transform concepts to a broader class of differential equations.
    • The generalized Hilbert transform H(mu) provides a novel tool for analyzing functions related to Yukawa potentials.