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We develop a scheme for obtaining the impedance, at any frequency, of a gradually tapered geometry of arbitrary cross-section containing a bunch of arbitrary profile travelling at the speed of light parallel to the axis of the taper. Coordinate-free expressions for Maxwell's equations are 2+2-split in a coordinate system adapted to the particle beam and the taper and, using an asymptotic expansion for a gradual taper, a coupled hierarchy of Poisson equations are obtained. Applications of the scheme are presented.
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