  In this paper, we develop a fast and accurate pseudospectral method to
approximate numerically the half Laplacian $(-\Delta)^{1/2}$ of a function on
$\mathbb{R}$, which is equivalent to the Hilbert transform of the derivative of
the function.
  The main ideas are as follows. Given a twice continuously differentiable
bounded function $u\in\mathcal C_b^2(\mathbb{R})$, we apply the change of
variable $x=L\cot(s)$, with $L>0$ and $s\in[0,\pi]$, which maps $\mathbb{R}$
into $[0,\pi]$, and denote $(-\Delta)_s^{1/2}u(x(s)) \equiv
(-\Delta)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of
$u(x(s))$, the problem is reduced to computing $(-\Delta)_s^{1/2}e^{iks} \equiv
(-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. On a previous work, we considered
the case with $k$ even for the more general power $\alpha/2$, with
$\alpha\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, we
express $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussian
hypergeometric function ${}_2F_1$, and also as a well-conditioned finite sum.
Then, we use a fast convolution result, that enable us to compute very
efficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$, for extremely
large values of $M$. This enables us to approximate $(-\Delta)_s^{1/2}u(x(s))$
in a fast and accurate way, especially when $u(x(s))$ is not periodic of period
$\pi$. As an application, we simulate a fractional Fisher's equation having
front solutions whose speed grows exponentially.
