  In the case of two-dimensional gradient index cavities designed by the
conformal transformation optics, we propose a boundary integral equation method
for the calculation of resonant mode functions by employing a fictitious space
which is reciprocally equivalent to the physical space. Using the Green's
function of the interior region of the uniform index cavity in the fictitious
space, resonant mode functions and their far-field distributions in the
physical space can be obtained. As a verification, resonant modes in
lima\c{c}on-shaped transformation cavities were calculated and mode patterns
and far-field intensity distributions were compared with those of the same
modes obtained from the finite element method.
