1. **State the problem:** Find the inverse Laplace transform of the function $$F(s) = \sqrt{s} - 1 + \ln \sqrt{s^2 - 1}$$. 2. **Understand the components:** The function involves terms $$\sqrt{s}$$, a constant $$-1$$, and a logarithmic term $$\ln \sqrt{s^2 - 1}$$. 3. **Recall Laplace inverse basics:** The inverse Laplace transform, denoted $$\mathcal{L}^{-1}\{F(s)\} = f(t)$$, converts a function in the complex frequency domain back to the time domain. 4. **Analyze each term separately:** - For $$\sqrt{s}$$, recall that $$\mathcal{L}\{t^{-3/2}\} = \frac{\sqrt{\pi}}{2} s^{1/2}$$ up to constants, but this is not a standard form. The inverse Laplace of $$s^{1/2}$$ is not elementary. - The constant $$-1$$ corresponds to $$-\delta(t)$$ in the time domain (impulse at zero). - For $$\ln \sqrt{s^2 - 1} = \frac{1}{2} \ln (s^2 - 1)$$, the inverse Laplace transform is complicated and not standard. 5. **Conclusion:** The given function is not a standard Laplace transform of elementary functions, and its inverse Laplace transform does not have a simple closed form. 6. **Recommendation:** To find the inverse Laplace transform of such expressions, consider numerical inversion methods or consult advanced integral transform tables. **Final answer:** The inverse Laplace transform of $$F(s) = \sqrt{s} - 1 + \ln \sqrt{s^2 - 1}$$ does not have a simple closed-form expression in elementary functions.