I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$
where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier transformation of a Dirac $\delta$ function. However, the integrand has two branch points, i.e. $k=0$ and $k=b$, here. Can this integral be calculated analytically, such as using residue theorem?