A method is presented to deal with the numerical evaluation of Kramers–Kronig transforms (the Hilbert transforms of even and odd functions on the positive real axis). The general Hilbert transform is also treated. The functions involved must be continuous on the integration interval with suitable asymptotic behavior for large values of the argument and must have an appropriate functional form in the vicinity of the singularity of the integrand of the transform. The approach is based on a specialized Gaussian quadrature technique that uses the weight function log x−1. This choice allows the region in the vicinity of the singularity to be swept into the quadrature weights and abscissa values. Application to the Lorentzian and Gaussian line profiles is discussed.
© 2002 Optical Society of America
Frederick W. King, "Efficient numerical approach to the evaluation of Kramers-Kronig transforms," J. Opt. Soc. Am. B 19, 2427-2436 (2002)