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The scintillation index for weak atmospheric turbulence is formulated for a partially coherent source (in space and time). We obtain results for two cases when the detector has a slow and a fast response time. Our solutions correctly reduce to the known scintillation index in the limit of coherent and incoherent sources. Transmitter aperture averaging and frequency averaging effects are shown as special cases of our results. The limitations of the source model used and the solution obtained for a fast detector are discussed.