Número complejo

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Aca seguimos los textos estándares, tal como Spiegel (1964) [1] o Levinson y Redheffer (1970). [2]


Un número con parte real e imaginaria, tal como


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=x+iy=re^{i\theta} } ,


donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=\sqrt{-1} } . [El símbolo j también se usa para indicar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}} y es una notación preferida por los ingenieros eléctricos, como el símbolo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i} se reserva con frecuencia para representar la corriente eléctrica]. El módulo o magnitud del número complejo anterior es Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r=\sqrt{x^2+y^2} } y el ángulo que indica su dirección respecto al eje real positivo

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta=\tan^{-1}{\frac{y}{x}} } .


La gráfica de una función o cantidad compleja (tal como el espectro de frecuencia) se muestra en la Figura C-10.



Entendiendo números complejos y funciones de variables complejas.

En el cálculo de las raíces de ecuaciones polinómicas, las cantidades aditivas que son escaladas por Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1} } aparecen a menudo. Por ejemplo, si consideramos una ecuación cuadrática general

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a z^2 + b z + c = 0 }

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a, b, c } son números reales. Las raíces de esta ecuación se pueden obtener a través de la fórmula cuadrática

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = \frac{ - b \pm \sqrt{ b^2 - 4 a c }}{ 2 a } .}

Acá vemos que cuando la cantidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b^2 - 4 ac < 0 } el resultado será


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = \frac{ - b \pm i \sqrt{ |4 a c - b^2|}}{ 2 a } = \frac{-b}{2a} \pm i \frac{\sqrt{|4 a c - b^2|}}{2a} }


donde usamos la convención matemática Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i = \sqrt{-1}. }

"El plano complejo z."


La representación polar y el plano de Argand

En general consideramos que un número complejo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y } donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x } es la "parte real y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y } es "la parte imaginaria" (llamada así, en un tiempo en que los matemáticos se sentían incómodos con cantidades que envolvían Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}.} )

Al matemático aficionado llamado Jean-Robert Argand se le atribuye de ser la primera persona en publicar la representación geométrica de los números complejos definida como un plano con eje horizontal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x } como el "eje real" y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle y} como "eje imaginario".

Si consideramos el ángulo Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta } como el ángulo en sentido opuesto a las manecillas del reloj desde el eje positivo real, entonces la representación polar natural resulta


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = x + i y = r( \cos(\theta) + i \sin(\theta) ) .}


Acá </math> where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{x^2 + y^2} = |z|. } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = |z|} se conoce como elmódulo de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z .}

El módulo se puede escribir como Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{(x + i y)(x -iy)} = \sqrt{z \overline{z}} = |z|^2 } donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overline{z} = x - i y } se conoce como "complejo conjugado" de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z.}


Otra representación polar está en las siguientes identidades

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z = r( \cos(\theta) + i \sin(\theta) ) = r e^{i \theta}, }

y

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \overline{z} = r( \cos(\theta) - i \sin(\theta) ) = r e^{-i \theta}, }

implicando que

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) \pm i \sin(\theta) = e^{\pm i \theta}. }


Comencemos por escribir formalmente las representaciones de la serie de Taylor de Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos (\theta) } y Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin (\theta) } y sumando la serie resultante


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos (\theta) = \sum_{k=0}^{\infty} \frac{(-1)^{k} \theta^{2k} }{ (2k)! } = \sum_{k=0}^{\infty} \frac{(i)^{2k} \theta^{2k} }{ (2k)! } = \sum_{k=0}^{\infty} \frac{( i \theta)^{2k} }{ (2k)! }, }


donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=0,1,2,3,... } . Podemos escribir la serie de Taylor de la forma Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin(\theta) }


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \sin (\theta) = i \sum_{l=0}^{\infty} \frac{(-1)^{l} \theta^{2l+1} }{ (2l+1)! } = \sum_{l=0}^{\infty} \frac{(i)^{2l+1} \theta^{2l+1} }{ (2l+1)! } = \sum_{l=0}^{\infty} \frac{( i \theta)^{2l+1} }{ (2l+1)! } , }


donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle l=0,1,2,3,...} . En los dos casos previos, se ha hecho uso libre de la identidad Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle -1 = i^2.}

La suma de estas dos series conlleva a la representación en serie de la función exponencial


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \cos(\theta) + i \sin (\theta) = \sum_{n=0}^{\infty} \frac{( i \theta)^{n} }{ n! } = e^{ i \theta }, }

donde Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n = 0,1,2,3,... } .

Probablemente no verá este argumento en un libro de texto ya que es un "argumento de plausibilidad" en lugar de una prueba, porque depende de tener la maquinaria de convergencia de la serie disponible.


Funciones de variables complejas

Si una función Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) } es una representación del campo de los números complejos al campo de los números complejos, entonces Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) } debe, por si misma, ser la suma de un función puramente real Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u(x,y) } y una función puramente imaginaria Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i v(x,y), } entonces,


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(z) = u(x,y) + i v(x,y). }


Cierta clase de función de valores complejos, conocida como Función Analítica es de particular importancia en la aplicación de funciones de valores complejos a los problemas de las ciencias físicas.


Referencias

<referencias />

  1. Spiegel, Murray R. "Theory and problems of complex variables, with an introduction to Conformal Mapping and its applications." Schaum's outline series (1964).
  2. Levinson, Norman, y Raymond M. Redheffer. "Complex variables." (1970), Holden-Day, New York.