The beamforming optimization in continuous aperture array (CAPA)-based multi-user communications is studied. In contrast to conventional spatially discrete antenna arrays, CAPAs can exploit the full spatial degrees of freedom (DoFs) by emitting information-bearing electromagnetic (EM) waves through continuous source current distributed across the aperture. Nevertheless, such an operation renders the beamforming optimization problem as a non-convex integral-based functional programming problem, which is challenging for conventional discrete optimization methods. A couple of low-complexity approaches are proposed to solve the functional programming problem. 1) Calculus of variations (CoV)-based approach: Closed-form structure of the optimal continuous source patterns are derived based on CoV, inspiring a low-complexity integral-free iterative algorithm for solving the functional programming problem. 2) Correlation-based zero-forcing (Corr-ZF) approach: Closed-form ZF source current patterns that completely eliminate the inter-user interference are derived based on the channel correlations. By using these patterns, the original functional programming problem is transformed to a simple power allocation problem, which can be solved using the classical water-filling approach with reduced complexity. Our numerical results validate the effectiveness of the proposed designs and reveal that: i) compared to the state-of-the-art Fourier-based discretization approach, the proposed CoV-based approach not only improves communication performance but also reduces computational complexity by up to hundreds of times for large CAPA apertures and high frequencies, and ii) the proposed Corr-ZF approach achieves asymptotically optimal performance compared to the CoV-based approach.