A new method to remove the eye-movement artifact from the electro-encephalogram (EEG) is described. It is based on complex regression analysis. Transfer of eye-movement activity to EEG can have frequency dependent amplitude and phase characteristics. The proposed method is suitable for handling such transfer because the regression formula is used in the frequency domain. The method is demonstrated with artificial signal-in-noise EOG (electro-oculogram) and EEG series. In the EEG noise an event related potential (ERP) is buried as a constant signal and in the EOG noise a changing EOG response (saccadic eye-movement) is simulated before adding the whole series to the EEG series. Also decreasing levels of transfer from EOG on EEG leads are simulated as the EOG artifact diminishes from the frontal to the occipital area. Because of the possible frequency dependent phase characteristics also a 24 msec time-shift of the EOG is simulated. The complex regression coefficient (P(jw)) of the EOG and EEG series is first calculated and then 'common' regression is removed from the EEG. To do this the estimated P(jw) is tested for significance with the F-statistic and significant EOG activity is subtracted in the frequency domain from the EEG on a single trial base. Then each EEG record is inversely transformed to the time domain. It was found that the complex P(jw)s as used in the subtraction formula, corrects very accurately the influence of the EOG artifact on EEG activity. Finally an application of the method in a real life situation is reported.