We present and compare two integral-equation formulations of the TE eigenvalue problem of a ridged circular waveguide. In the E-formulation, an integral equation for the tangential electric field Eø. is derived and solved by the moment method using basis functions which include the edge conditions. The H-formulation relies on an integral equation for the axial magnetic field Hz, which is also solved by the moment method and basis functions with edge conditions. Spurious roots are encountered in the E-formulation whereas the H-Formulation is free from spurious roots but requires larger matrices. Results from both formulations are compared with previously published data and analytic results for limiting cases, excellent agreement is demonstrated.