Álgebra y Geometría Analítica

Contenidos de Álgebra para UTN-FRBA

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Regiones del plano complejo

A continuación veremos una serie de ejemplos que muestran cómo se puede graficar una región del plano complejo que cumple con ciertas condiciones.

Ejemplo 1

Hallar la región del plano complejo determinada por:
\[\left\{ {z \in \mathbb{C}\;:\;\;\;\left| {z + 2} \right| \le 1} \right\}\]

Resolución

Si \(z = x + yi\) , resulta:

\[\left| {x + yi + 2} \right| \le 1\]

\[\left| {\left( {x + 2} \right) + yi\;} \right| \le 1\]

\[\sqrt {{{\left( {x + 2} \right)}^2} + {y^2}} \le 1\]

Observación importante: La unidad imaginaria no interviene en el cálculo del módulo:

\[\left| {a + bi} \right| = \sqrt {{a^2} + {b^2}} \]

Elevando al cuadrado ambos miembros:
\[{\left( {x + 2} \right)^2} + {y^2} \le 1\]

Podemos sustituir el símbolo de \( \le \) por el de \( = \) para obtener la frontera o borde de la región:
\[{\left( {x + 2} \right)^2} + {y^2} = 1\]

Ésta es la ecuación de una circunferencia con centro en \(\left( { – 2,0} \right)\) y radio 1.

Veamos una gráfica de la región:

Región del plano complejo

Son todos los puntos que están a distancia menor o igual que 1 del punto (-2,0)

Ejemplo 2

Hallar la región del plano complejo determinada por:
\[\left\{ {z \in \mathbb{C}\;\;:\;\;\;\left| z \right| \le 2\;\;,\;\;\frac{\pi }{2} \le \arg \left( z \right) \le \frac{3}{4}\pi } \right\}\]

Resolución

Esta región está definida utilizando la expresión trigonométrica de un número complejo. \(\left| z \right| \le 2\) quiere decir que el módulo debe ser menor o igual a 2, y la otra condición establece que el argumento está entre \(\pi /2\) y \(3/4\pi \).

Luego:

región del plano complejo

Observación: \(z = \left( {0,0} \right)\) no está incluido, porque no tiene argumento.

Ejemplo 3

Hallar la región del plano complejo determinada por:

    \(\left\{ {z \in \mathbb{C}\;\;:\;\;\;\left| {z + i} \right| \le 2\;\;,\;\;\frac{\pi }{2} \le \arg \left( z \right) \le \frac{3}{4}\pi } \right\}\)

Resolución

Esta región está definida utilizando la expresión trigonométrica de un número complejo. \(\left| {z + i} \right| \le 2\)representa al círculo con centro en \( – i\) de radio 2. La otra condición establece que el argumento está comprendido entre \(\pi /2\) y \(3/4\pi \).

Luego la intersección entre ambas regiones es:

Noten que \(z = 0\) no está incluido porque no tiene argumento.

Observación: un error muy común es indicar el ángulo dado por \(\frac{\pi }{2} \le \arg \left( z \right) \le \frac{3}{4}\pi \) con vértice en el centro de la circunferencia:

Ejemplo 4

\[\left\{ {z \in \mathbb{C}\;:\;\;\;\left| {z + i} \right| = \left| {z – 1} \right|\;\;,\;\;Re\left( z \right) > 1} \right\}\]

Resolución
Razonamiento analítico

\[\left| {x + iy + i} \right| = \left| {x + iy – 1} \right|\]

\[\left| {\;x + i\left( {y + 1} \right)} \right| = \left| {\;\left( {x – 1} \right) + iy\;} \right|\]

Calculamos el módulo de cada complejo:
\[\sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} = \sqrt {{{\left( {x – 1} \right)}^2} + {y^2}} \;\;\;\;\;,\;\;\;\;\;x > 1\]

\[{x^2} + {\left( {y + 1} \right)^2} = {\left( {x – 1} \right)^2} + {y^2}\;\;\;\;,\;\;\;\;\;x > 1\]

\[{x^2} + {y^2} + 2y + 1 = {x^2} – 2x + 1 + {y^2}\;\;\;\;\;,\;\;\;\;\;x > 1\]

\[2y + 1 = – 2x + 1\;\;\;\;\;\;,\;\;\;\;\;x > 1\]

\[y = – x\;\;\;,\;\;x > 1\]

Interpretación geométrica

Si \(a,b \in \mathbb{R}\) entonces \(\left| {a – b} \right|\) representa la distancia entre \(a\) y \(b\).

Análogamente se puede definir la distancia en \(\mathbb{C}\). Sean \({z_1},{z_2} \in \mathbb{C}\) entonces \(\left| {{z_1} – {z_2}} \right|\) es la distancia entre \({z_1}\) y \({z_2}\).

Luego \(\left| {z + i} \right| = \left| {z – \left( { – i} \right)} \right|\)representa la distancia entre \(z\) y \( – i\).

Y \(\left| {z – 1} \right|\) representa la distancia entre \(z\) y \(1\).

La ecuación\(\left| {z + i} \right| = \left| {z – 1} \right|\)caracteriza al conjunto de números complejos que están a igual distancia de\(\left( { – i} \right)\) que de \(1\).

Traducido a \({\mathbb{R}^2}\), se trata del conjunto de puntos del plano que equidistan de \(A\left( {0,\; – 1} \right)\) y \(B\left( {1,0} \right)\), o sea, es la mediatriz del segmento \(AB\). Considerando la restricción \(Re\left( z \right) > 1\), se obtiene la gráfica anterior.

Ejemplo 5

Hallar la región del plano complejo determinada por:
\[Im\left( {z – i} \right) = Re\left( {z + 1} \right)\]

Resolución

Escribiremos al número complejo \(z\) como \(x + yi\):
\[Im\left( {x + iy – i} \right) = Re\left( {x + iy + 1} \right)\]
\[y – 1 = x + 1\]
\[y = x + 2\]

Veamos una gráfica de la región (que es sencillamente la recta \(y = x + 2)\):

Ejemplo 6

Hallar la región del plano complejo determinada por:
\[\left\{ {\begin{array}{*{20}{c}}{\left| {z – i} \right| + \left| {z + i} \right| = 4}\\{0 \le Arg\left( z \right) \le \pi }\end{array}} \right.\]

Resolución

\[\left| {x + yi – i} \right| + \left| {x + yi + i} \right| = 4\]
\[\sqrt {{x^2} + {{\left( {y – 1} \right)}^2}} + \sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} = 4\]
\[\sqrt {{x^2} + {{\left( {y – 1} \right)}^2}} = 4 – \sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} \]

Elevamos al cuadrado a ambos miembros:
\[{x^2} + {\left( {y – 1} \right)^2} = {4^2} + {x^2} + {\left( {y + 1} \right)^2} – 8\sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} \]
\[{y^2} – 2y + 1 = {4^2} + {y^2} + 2y + 1 – 8\sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} \]

\[ – 4y – 16 = – 8\sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} \]

\[\frac{{y + 4}}{2} = \sqrt {{x^2} + {{\left( {y + 1} \right)}^2}} \]

Volvemos a elevar al cuadrado a ambos miembros:
\[{\frac{{\left( {y + 4} \right)}}{4}^2} = {x^2} + {\left( {y + 1} \right)^2}\]

\[{\left( {y + 4} \right)^2} = 4{x^2} + 4{\left( {y + 1} \right)^2}\]

\[{y^2} + 8y + 16 = 4{x^2} + 4{y^2} + 8y + 4\]

\[0 = 4{x^2} + 3{y^2} – 12\]

\[12 = 4{x^2} + 3{y^2}\}

\[\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 1\]

Se trata de la ecuación de una elipse con eje focal vertical y centrada en \(\left( {0,0} \right)\):

Pero también debe ser:
\[0 \le Arg\left( z \right) \le \pi \]

Luego no es toda la elipse sino aquel arco que cumple con que el argumento está comprendido entre 0 y \(\pi \):

Interpretación geométrica

\[\left\{ {\begin{array}{*{20}{c}}{\left| {z – i} \right| + \left| {z + i} \right| = 4}\\{0 \le Arg\left( z \right) \le \pi }\end{array}} \right.\]

\(\left| {z – i} \right|\): representa la distancia de\(\;\;z\) a \(i\)

\(\left| {z + i} \right|\): representa la distancia de \(\;z\) a -\(i\)

Es decir que la suma de esas distancias debe ser constante, e igual a 4.

Traducido a \({\mathbb{R}^2}\), es el conjunto de puntos del plano tales que la suma de sus distancias a dos puntos fijos: \(\left( {0,1} \right)\) y \(\left( {0,\; – 1} \right)\) es igual a 4. De acuerdo con lo que hemos visto en la unidad de cónicas, se trata de la elipse de focos \(\left( {0,1} \right)\) y \(\left( {0,\; – 1} \right)\) y semieje mayor \(a\; = \;2\).

Para hallar el semieje menor aplicamos la relación: \({a^2} = {b^2} + {c^2}\) , a partir de la cual se obtiene \(b = \sqrt 3 \) .

Ejercicio para el lector 5

Hallar analíticamente y graficar la región del plano complejo definida por:

\[\{ z \in C:\;\;\left| {z – 2i} \right| < 2\;\;,\;\;Im\left( {{z^2}} \right) \le 0\;\;,\;\;\;\frac{\pi }{4} \le \arg \left( z \right) \le \frac{{3\pi }}{4}\;\;\} \]

Videos relacionados con regiones del plano complejo

 

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Comentarios

  1. Nacho dice

    Mil gracias por este contenido, Federico e Isabel. Sin ésta página no creo que podría haber aprobado! Piensan hacer lo mismo con otras materias?

    Se nota todo el trabajo que le pusieron, y se aprecia MUCHÍSIMO! Son unos genios.

  2. Isabel dice

    Hola Nacho!
    ¡Muchas gracias por tu comentario tan gratificante! ¡Qué bueno que los materiales te hayan sido útiles!
    Respecto de otras materias, sé que hay proyectos en curso pero los responsables de concretarlos son los profesores de esas materias.
    ¡Felicitaciones por haber aprobado!
    Saludos y felices fiestas!
    Isabel

  5. Isabel Pustilnik y Federico Gómez dice

    Sergio, gracias por comentar. Los gráficos están hechos con GeoGebra (es libre y gratuito).

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