  We present a systematic analysis of the polarised and unpolarised processes
$e^+ ~e^- \rightarrow e^+ ~e^- X$ in the deep inelastic limit and study the
effects of target photon mass (virtuality) on the photon structure functions.
The effect of target photon virtuality manifests as new singly polarised
structure functions and also alters the physical interpretation of the
unpolarised structure functions. The physical interpretation of these structure
functions in terms of hadronic components is studied using the free field
analysis. We also retrieve the real photon results in the limit the virtuality
goes to zero. Assuming factorisation of the photon structure tensor, the
relevant QCD corrections to the various twist two structure functions are
evaluated.
