Norm of function $h \in L^2(\mathbb{R}^3)$

Question

Let $h\colon \mathbb{R}^3 \to \mathbb{C}$, $$h(x)=\frac{1}{|x|(|x|^2+1)^{1/2}}$$

For which $p \text{ does } h \in L^p(\mathbb{R}^3)?$

In case $h \in L^2(\mathbb{R}^3)$ what is the corresponding norm?

Can you can use the residues method?

Answer 1

The idea is to exploit radial symmetry (see here for a formal justification):

\begin{align}\|h\|_{L^p(\mathbb{R}^3)}^p&=\int_{\mathbb{R}^3}|h(x)|^pdx=\int_0^{\infty}\int_{|x|=r}|h(x)|^pdx dr=\int_0^{\infty}4\pi r^2\frac{1}{r^p(r^2+1)^{p/2}}dr=\\ &=4\pi\int_0^{\infty}\frac{r^{2-p}}{(r^2+1)^{p/2}}dr\end{align}

and now your problem is reduced to one-dimensional calculus.