Computing the Hilbert Transform of the Generalized Laguerre and Hermite Weight Functions

@inproceedings{Gautschi2000ComputingTH,
  title={Computing the Hilbert Transform of the Generalized Laguerre and Hermite Weight Functions},
  author={Walter Gautschi and J{\"o}rg Waldvogel},
  year={2000}
}
We give explicit formulae for the Hilbert transform ∫ R − w(t)dt/(t − x), where w is either the generalized Laguerre weight function w(t) = 0 if t ≤ 0, w(t) = tαe−t if 0 < t < ∞, and α > −1, x > 0, or the Hermite weight function w(t) = e−t 2 , −∞ < t <∞, and −∞ < x < ∞. Furthermore, several numerical evaluation schemes are discussed, based on various representations of the objects under consideration. In this connection we study the numerical stability of the three-term recurrence relation… CONTINUE READING

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