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We develop a method to synthesize any partially coherent source (PCS) with a genuine cross-spectral density (CSD) function using complex transmittance screens. Prior work concerning PCS synthesis with complex transmittance screens has focused on generating Schell-model (uniformly correlated) sources. Here, using the necessary and sufficient condition for a genuine CSD function, we derive an expression, in the form of a superposition integral, that produces stochastic complex screendoi:10.1364/josaa.381772 pmid:32118906 fatcat:mr2nejbyyvfwrlqllm5beee2fm