In this paper we present a new method for computing the pseudospectra of delay differential equations (DDEs) with fixed finite delay. This provides information on the sensitivity of eigenvalues under arbitrary perturbations of a given size, and hence insight into how stability may change under variation of parameters. We also investigate how different weighted perturbations applied to the individual matrices of the delayed eigenvalue problem affect the pseudospectrum. Furthermore, we compute pseudospectra of the infinitesimal generator of the DDE, from which a lower bounds on the maximum transient growth can be inferred. To illustrate our method, we consider a DDE modelling a semiconductor laser subject to external feedback.

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