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| In [[mathematical analysis]], the '''Cauchy index''' is an [[integer]] associated to a real [[rational function]] over an [[Interval (mathematics)|interval]]. By the [[Routh–Hurwitz theorem]], we have the following interpretation: the Cauchy index of
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| :''r''(''x'') = ''p''(''x'')/''q''(''x'')
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| 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
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| :''f''(''iy'') = ''q''(''y'') + ''ip''(''y'').
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| We must also assume that ''p'' has degree less than the degree of ''q''.
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| ==Definition==
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| * The '''Cauchy index''' was first defined for a pole ''s'' of the rational function ''r'' by [[Augustin Louis Cauchy]] in 1837 using [[one-sided limit]]s as:
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| :<math> I_sr = \begin{cases}
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| +1, & \text{if } \displaystyle\lim_{x\uparrow s}r(x)=-\infty \;\land\; \lim_{x\downarrow s}r(x)=+\infty, \\
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| -1, & \text{if } \displaystyle\lim_{x\uparrow s}r(x)=+\infty \;\land\; \lim_{x\downarrow s}r(x)=-\infty, \\
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| 0, & \text{otherwise.}
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| \end{cases}</math>
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| * A generalization over the compact interval [''a'',''b''] is direct (when neither ''a'' nor ''b'' are poles of ''r''(''x'')): it is the sum of the Cauchy indices <math>I_s</math> of ''r'' for each ''s'' located in the interval. We usually denote it by <math>I_a^br</math>.
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| * We can then generalize to intervals of type <math>[-\infty,+\infty]</math> since the number of poles of ''r'' is a finite number (by taking the limit of the Cauchy index over [''a'',''b''] for ''a'' and ''b'' going to infinity).
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| ==Examples==
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| [[Image:cauchyindex.png|thumb|300px|A rational function]]
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| * Consider the rational function:
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| :<math>r(x)=\frac{4x^3 -3x}{16x^5 -20x^3 +5x}=\frac{p(x)}{q(x)}.</math> | |
| We recognize in ''p''(''x'') and ''q''(''x'') respectively the [[Chebyshev polynomials]] of degree 3 and 5. Therefore ''r''(''x'') has poles <math>x_1=0.9511</math>, <math>x_2=0.5878</math>, <math>x_3=0</math>, <math>x_4=-0.5878</math> and <math>x_5=-0.9511</math>, i.e. <math>x_j=\cos((2i-1)\pi/2n)</math> for <math>j = 1,...,5</math>. We can see on the picture that <math>I_{x_1}r=I_{x_2}r=1</math> and <math>I_{x_4}r=I_{x_5}r=-1</math>. For the pole in zero, we have <math>I_{x_3}r=0</math> since the left and right limits are equal (which is because ''p''(''x'') also has a root in zero).
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| We conclude that <math>I_{-1}^1r=0=I_{-\infty}^{+\infty}r</math> since ''q''(''x'') has only five roots, all in [−1,1]. We cannot use here the Routh–Hurwitz theorem as each complex polynomial with ''f''(''iy'') = ''q''(''y'') + ''ip''(''y'') has a zero on the [[imaginary number|imaginary line]] (namely at the origin).
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| ==External links==
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| * [http://deslab.mit.edu/DesignLab/itango/multi/sld008.htm The Cauchy Index]
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| [[Category:Mathematical analysis]]
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