Convergence of the Continued Fraction for the Square Root Function
, by Bert Jacobs and Francis Muir
The square root function can be approximated with a continued fraction for use in wave equation migration algorithms. This continued fraction can be generated by a recursion. If no dissipation other than dip filtering is employed then the recursion conberges in the propogating region and on the boundary between the propogating and evanescent regimes. The recursion diverges in the evanescent zone and at zero temporal frequency. Depending on the starting point convergence may also occur along a pair of radial lines in the fk-plane.
The evanescent zone disappears when the wave operator is causal and has a spectrum with a strictly-positive real part. The continued fraction in this case will converge essentially everywhere.
STANFORD