The inverse transform of $\sqrt{s}$ doesn't exist, because the integral representing it,

$$ \int\limits_{\gamma -i\infty }^{\gamma +i\infty }\sqrt{s}e^{st}\mathrm{d}s, $$

diverges as $s \rightarrow \infty$. People typically don't notice this, and just go ahead with "evaluating" this integral (by closing the contour, etc.) -- and end up with a result which doesn't make sense (as the resulting function doesn't have a forward transform).

The inverse transform of $\sqrt{s}$ doesn't exist, because the integral representing it,

$$ \int\limits_{\gamma -i\infty }^{\gamma +i\infty }\sqrt{s}e^{st}\mathrm{d}s, $$

diverges as $s \rightarrow \gamma \pm i\infty$. People typically don't notice this, and just go ahead with "evaluating" this integral (by closing the contour, etc.) -- and end up with a result which doesn't make sense (as the resulting function doesn't have a forward transform).

The inverse transform of $\sqrt{s}$ doesn't exist, because the integral representing it,

$$ \int\limits_{\gamma -i\infty }^{\gamma +i\infty }\sqrt{s}e^{st}\mathrm{d}s, $$

diverges as $s \rightarrow \infty$. People typically don't notice this, and just go ahead with "evaluating" this integral (by closing the contour, etc.) -- and end up with a result which doesn't make sense (as the resulting function of doesn't have a forward transform).

The inverse transform of $\sqrt{s}$ doesn't exist, because the integral representing it,

$$ \int\limits_{\gamma -i\infty }^{\gamma +i\infty }\sqrt{s}e^{st}\mathrm{d}s, $$

diverges as $s \rightarrow \infty$. People typically don't notice this, and just go ahead with "evaluating" this integral (by closing the contour, etc.) -- and end up with a result which doesn't make sense (as the resulting function doesn't have a forward transform).