9.1: Polos y ceros
- Page ID
- 109864
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Te recordamos la siguiente terminología: Supongamos que\(f(z)\) es analítico en\(z_0\) y
\[f(z) = a_n (z - z_0)^n + a_{n + 1} (z - z_0)^{n + 1} + \ ...,\]
con\(a_n \ne 0\). Entonces decimos que\(f\) tiene un cero de orden\(n\) en\(z_0\). Si\(n = 1\) decimos\(z_0\) es un simple cero.
Supongamos que\(f\) tiene una sigularidad aislada en\(z_0\) y serie Laurent
\[f(z) = \dfrac{b_n}{(z - z_0)^n} + \dfrac{b_{n - 1}}{(z - z_0)^{n - 1}} + \ ... + \dfrac{b_1}{z - z_0} + a_0 + a_1 (z - z_0) + \ ...\]
que converge en\(0 < |z - z_0| < R\) y con\(b_n \ne 0\). Entonces decimos que\(f\) tiene un polo de orden\(n\) en\(z_0\). Si\(n = 1\) decimos\(z_0\) es un simple polo.
Hay varios ejemplos en las notas del Tema 8. Aquí hay uno más
\[f(z) = \dfrac{z + 1}{z^3 (z^2 + 1)} \nonumber\]
tiene singularidades aisladas en\(z = 0\),\(\pm i\) y un cero en\(z = -1\). Mostraremos que\(z = 0\) es un polo de orden 3,\(z = \pm i\) son polos de orden 1 y\(z = -1\) es un cero de orden 1. El estilo de argumento es el mismo en cada caso.
En\(z = 0\):
\[f(z) = \dfrac{1}{z^3} \cdot \dfrac{z + 1}{z^2 + 1}. \nonumber\]
Llama al segundo factor\(g(z)\). Ya que\(g(z)\) es analítico en\(z = 0\) y\(g(0) = 1\), tiene una serie Taylor
\[g(z) = \dfrac{z + 1}{z^2 + 1} = 1 + a_1 z + a_2 z^2 + \ ... \nonumber\]
Por lo tanto
\[f(z) = \dfrac{1}{z^3} + \dfrac{a_1}{z^2} + \dfrac{a_2}{z} + \ ... \nonumber\]
Esto muestra\(z = 0\) es un polo de orden 3.
En\(z = i\):\(f(z) = \dfrac{1}{z - i} \cdot \dfrac{z + 1}{z^3 (z + i)}\). Llama al segundo factor\(g(z)\). Ya que\(g(z)\) es analítico en\(z = i\), tiene una serie Taylor
\[g(z) = \dfrac{z + 1}{z^3 (z + i)} = a_0 + a_1 (z - i) + a_2 (z - i)^2 + \ ... \nonumber\]
donde\(a_0 = g(i) \ne 0\). Por lo tanto,
\[f(z) = \dfrac{a_0}{z - i} + a_1 + a_2 (z - i) + \ ... \nonumber\]
Este espectáculo\(z = i\) es un polo de orden 1.
Los argumentos a favor\(z = -i\) y\(z = -1\) son similares.