My question is that, what is the inverse Laplace transform of $\sqrt{1+a/s}$ where $a \gt 0$?
I tried to find solution in the Integral and Series books, but failed.
2026-04-24 02:25:13.1776997513
What is the inverse Laplace transform of $\sqrt{1+a/s}$?
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The "solution from a book" uses modified Bessel functions: $$\mathcal{L}^{-1}\left[\sqrt{1+2a/s}-1\right](t)=ae^{-at}\big(I_0(at)+I_1(at)\big)$$ (easily obtained from the LT of $I_\alpha$; should I find a reference to the latter?..).
Knowing the ILT of $1$ (the first entry of this table), you're done.