Complex polynomials representation of solutions to the antenna synthesis problem by the power pattern

The nonlinear integral equation of Hammerstein type arising in the antenna synthesis problem according to the given power pattern is investigated. This equation contains the nonlinearity of the third degree in the integrant. Its solutions are presented by a real positive function (amplitude pattern) and a polynomial with complex zeros (whose phase coincides with the phase pattern). The both amplitude pattern and zeros of the polynomials are found from a set of equations, one of which is the nonlinear integral equation, and the rest (as many as twice the polynomial degree) are the transcendental ones. Numerical results for a concrete problem are described.