The most general solution to the cylindrical tearing mode equation, (3.60), in the outer region can now be writtenwhere the real functions and are specified in Sections 3.8 and 3.9, respectively. Moreover, the real function is a solution of that satisfies
In general, is continuous across the field-coil (in accordance with Maxwell's equations), whereas is discontinuous. The discontinuity in is caused by the helical current flowing in the field-coil. The complex quantitydetermines the amplitude and phase of the perturbed magnetic flux at the field-coil. The complex quantity parameterizes the amplitude and phase of the helical current sheet flowing in the field-coil. It follows from Equations (3.37), (3.38), (3.110), and (3.111) that
Simultaneously matching the outer solution, (3.112), across the rational surface, the resistive wall, and the field-coil, we obtainHere, where use has been made of Equations (3.81), (3.118), and (3.119).