An analogue of complex variable theory in the plane is obtained by considering a three-dimensional distribution basic spinors. These satisfy the invariant linear differential equation denoted by curl(W)=0, which plays the role of the Cauchy-Riemann equations. Spinor harmonics and associated spinor harmonics are evaluated. The analogues of Taylor series and Laurent series are obtained. Analytic continuation is discussed, and this leads to the analogue of a Riemann surface in three dimensions. These spinor fields are equivalent to the space-time spinor fields which describe elementary particles, the case considered here corresponding to zero time derivative.