Numerical work indicates that resistive instability may be the dominant mod
e of instability in the Earth's outer core for realistic core parameter reg
imes. In this paper, we assume that the Elsasser number is large in order t
o obtain an asymptotic analysis of resistive instability in an electrically
conducting fluid confined to a rotating cylindrical shell of infinite exte
nt in the axial direction. The dimensionless equations of motion are linear
ized about an ambient magnetic field which is purely azimuthal and depends
only on the cylindrical radial variable. Applying the theory of ordinary di
fferential equations with a large parameter, we obtain an asymptotic approx
imation to the solution. Relatively simple analytic expressions for the com
plex frequencies are obtained by applying the boundary conditions for insul
ating boundaries at the cylindrical sidewalls and then assuming that the am
bient magnetic field vanishes at one or both of those sidewalls. The result
s appear to be consistent with previous numerical work.