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Rouche 's theorem , named after Eugene Rouche , states that for any two complex-valued functions f and g holomorphic inside some region K with closed contour , if  |  g -LRB- z -RRB-  |  <  |  f -LRB- z -RRB-  |  on , then f and f + g have the same number of zeros inside K , where each zero is counted as many times as its multiplicity .. Eugene Rouche. Eugene Rouche. complex. complex number. functions. function ( mathematics ). holomorphic. Holomorphic function. multiplicity. Multiplicity ( mathematics ). This theorem assumes that the contour is simple , that is , without self-intersections .. Rouche 's theorem is an easy consequence of a stronger symmetric Rouche 's theorem described below .