I made a lamp rig and but whether I use the bulb or a circle in front of the bulb as an emissive surface it casts a shadow in the middle of the lit area. What might be causing this?
The issue is not with the light itself, but with the Velvet material used for the backdrop.
If you make that material just a diffuse material you'll see that the light works perfectly...
(I switched the window emitter in the scene just to show the effect clearly)
Whatever effect you are after with the velvet you might need to achieve through different means.
Also if you are interested in a more realistic light bulb (no need for a different mesh as emitter) you can read: Glass render problem and Light-bulb Filaments: Brightness and Internal Reflections in Cycles