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Stationary complex-valued functionals are discussed and seen to have unique stationary points if they are analytic, which corresponds to analyticity of functions in complex-valued function theory. As an application, the problem of wave propagation in a circular waveguide with an impedance boundary and the corresponding two-dimensional resonator are discussed. It is seen that the variational method for nonstandard eigenvalue problems, introduced a few years ago, can be generalised to complex-valued functionals if the functional is analytic. The method can be applied to result in simple analytic formulas for the waveguide with impedance boundaries.