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In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index of :''r''(''x'') = ''p''(''x'')/''q''(''x'') over the real line is the difference between the number of roots of ''f''(''z'') located in the right half-plane and those located in the left half-plane. The complex polynomial ''f''(''z'') is such that :''f''(''iy'') = ''q''(''y'') + ''ip''(''y''). We must also assume that ''p'' has degree less than the degree of ''q''. ==Definition== * The Cauchy index was first defined for a pole ''s'' of the rational function ''r'' by Augustin Louis Cauchy in 1837 using one-sided limits as: : * A generalization over the compact interval () is direct (when neither ''a'' nor ''b'' are poles of ''r''(''x'')): it is the sum of the Cauchy indices of ''r'' for each ''s'' located in the interval. We usually denote it by . * We can then generalize to intervals of type since the number of poles of ''r'' is a finite number (by taking the limit of the Cauchy index over () for ''a'' and ''b'' going to infinity). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cauchy index」の詳細全文を読む スポンサード リンク
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