On increasing stability in the two dimensional inverse source scattering problem with many frequencies

In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\mathrm{\Omega }$ with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.