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I need the inverse Laplace transform $\mathscr{L}^{-1}$ or a nice upper bound on $\mathscr{L}^{-1}$ for the following function: $$f(y)=\frac{\left(\sum_i \frac{u_i}{y-a_i}\right)^2}{1-\sum_i u_i \frac{u_i}{y-a_i}}$$

Any advice?

  • This comes out of rank-1 correction to matrix exponential of diagonal matrix.

  • $\mathscr{L}^{-1}f=\langle\exp(t (A + B)) - \exp(t A), \mathbf{1}\rangle$ where $\langle M, \mathbf{1}\rangle=\sum_{ij}M_{ij}$, $A$ diagonal, $B=uu'$ is rank-1

  • Fedya gave a nice numeric algorithm to compute this, now I need higher-level understanding such as:

  • if $u_i=i^{-1.12}$, and $a_i=-2u_i(1-u_i)$, how fast does $\mathscr{L}^{-1}f$ decay? Reason for $-1.12$ is Figure 9 here.

  • also asked on math.SE a few days ago. A user on that forum pointed out that roots of $f(y)$ appear to be real

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  • $\begingroup$ How about we use the inverse-LT contour formula and residue since it seems that you have exact information on the location of the poles? $\endgroup$
    – Thomas Kojar
    Commented Mar 28, 2023 at 2:10
  • $\begingroup$ I can get explicit formula for specific value of $a$ by plugging it into Mathematica. I'm rather looking for nice bounds, ie, $g(t)<t^{-0.25}$ when $u_i=i^{-1.12}$ and $a_i=-2u_i(1-u_i)$ $\endgroup$
    – Yaroslav Bulatov
    Commented Mar 28, 2023 at 2:14
  • $\begingroup$ the contour-formula should give expressions for general $a$ too. From the residue-expressions the bounds might be easy to get. $\endgroup$
    – Thomas Kojar
    Commented Mar 28, 2023 at 2:15

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