DEPARTAMENTO DEPARTAMENTO ENERGIA Y MECANICA
CARRERA INGENIERÍA MECATRONICA CUARTO NIVEL “CUADERNO DE MATEMÁTICA SUPERIOR”
ING. JHONY BASTIDAS
INTEGRANTES:
Latacunga ABRIL /2012
1
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UNIDAD I
ANALISIS COMPLEJO
1.1 Funciones y Sucesiones Complejas 1.2 Límites y Continuidad 1.3 Derivabilidad 1.4 Integral Compleja y el TFC 1.5 Series de Potencia Complejas. 1.6 Teorema de Cauchy – Goursat y Formula Integral de Cauchy. 1.7 Series de Laurent, Teorema de los Residuos UNIDAD II
ANALISIS DE FOURIER
2.1 Espacios Euclidanos y proceso de Gram – Schmidt 2.2 Bases Hilbertianas 2.3 El Espacio L2 2.4 Bases Hilbertianas de L 2 2.5 Base Clásica de Fourier 2.6 Transformada de Fourier UNIDAD III
ECUACIONES DIFERENCIALES DIFERENCIALES PARCIALES
3.1 Introducción a los modelos matemáticos 3.2 EDP Clásicas ecuaciones de onda, calor y Laplace 3.3 Método de Separación de Variable 3.4 Funciones Generalizadas 3.5 Aplicación 3.5 Aplicación de la transformada transformada de de Fourier y Laplace para para resolver resolver un EDP
BIBLIOGRAFIA
Matemática Superior para Ingeniería con ayuda de Máximo / Mayorga I. / 2011 / Español Maxima by Example / Edwin L. Wodlet / 2009 / Inglés California Uneversity State Long Beach. Maxima and the Calculus / Leon Q. Brin / 2009 / Inglés Southesm University CT State.
Matemática Avanzada para Ingeniería / Sexta Edición Peter 0’ Heil / 2009 / Español
Cencage Lerning. Análisis Matemática Avanzada con aplicaciones a Ingeniería y Ciencias / Reddy I y Rsmussen / 1992 / Español Limusa. hppt://forja rediris.es/docman/new.php/ rediris. es/docman/new.php/209/356/guía_wxmaxima.p 209/356/guía_wxmaxima.pdf//Máxima df//Máxima software libre en el aula de matemáticas//Opción: Lectura Obligatoria.
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UNIDAD I NUMEROS COMPLEJOS x+bi o x+b, x+b, en donde x y y son números reales e i2 Un numero complejo es de la forma x+bi o = 1. 1. La aritmética de los números complejos está definida c+id esto definida por igualdad a+ib = c+id esto c y b = d. d. sucede cuando exactamente a = c y Suma:
Multiplicación:
En la multiplicación de números complejos se procede exactamente como se lo hace con polinomios. El número real a se llama la parte real de a+ib y a+ib y es denotado por a = Re (a+ib). (a+ib). El número real b es la parte imaginaria y se denota por b = Im (a+ib). (a+ib). EJEMPLO:
Tanto la parte real como la imaginaria de cualquier número complejo son números reales. Si se tiene el sistema de los números complejos como una extensión del sistema de los números reales en el sentido que todo número real a es el numero complejo a +0i +0i.. Esta extensión de los números reales a los complejos tienen consecuencias profundas, tanto para el álgebra como para el análisis matemático por ejemplo, la ecuación binomial x2+1 = 0 no tiene solución real, pero tiene dos soluciones complejas i y – i. En general el teorema fundamental de algebra establece que todo polinomio de grado positivo n, con coeficientes complejos, tiene exactamente n raíces en los números complejos, contando las raíces repetidas, esto significa que no necesita extender los números complejos para encontrar las raíces de los polinomios con coeficientes complejos, como sucede con los números reales, para encontrar las raíces de un polinomio simple tal como x2+1. +1.
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zw=wz z + (w + u) = (z + w) + u z (w u) = (z w) u z (w + u) = z w + z u
Conmutativa de la Multiplicación Multiplicac ión Asociativa de la Suma Asociativa de la Multiplicación Multiplicac ión Distributiva
EL PLANO COMPLEJO Los números complejos admiten dos interpretaciones geométricas naturales, la primera identifica al número complejo (a + bi) con el punto par ordenado (a, b) en b) en el punto, como se muestra en la figura. Eje Imaginario
.
(a, b)-(a+bi)
Eje Real Esta interpretación cada número real a o a +0i +0i está identificado con el punto par ordenado (a, 0) en 0) en el eje horizontal, el cual es por tanto llamado el eje real. Un numero 0 + bi bi o solo bi bi se llama un numero imaginario puro y está asociado con el punto par ordenado (0, b) b) en el eje vertical. Este eje se llama el eje imaginario. Debido a esta correspondencia, entre los números complejos y los números en el plano, serán referidos en el plano xy como xy como el plano complejo. La segunda interpretación geométrica de los números complejos es en términos de vectores. El número complejo z = a + bi o bi o el punto par ordenado (a, b) puede pensarse como un vector ai + bj bj en el plano el cual a su vez es representado como una flecha desde el origen al punto (a, b). b). y
(a, b) (a+bi)
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La primera componente de este vector es Re (z) y la segunda componente de este (z), en esta interpretación, la definición de suma de números complejos es vector es Im (z), equivalente a la ley del paralelogramo para la suma de vectores, ya que dos vectores se suman, sumando sus componentes respectivas. y (a+c) + i(b+d) (a+bi)
(c+di) x
MAGNITUD La magnitud de a + bi es denotada por la + bi l y está definida por la + bi I = , por supuesto, la magnitud de cero es cero si z = a + bi es un numero complejo distinto de cero, entonces la magnitud de z es la distancia del origen al punto (a, b). b). Alternativamente, I z I es I es la longitud de ai + bj representado bj representado a..
La magnitud de un numero complejo también se llama modulo. y (a, b) (a+bi) Ia+biI
x CONJUGADA
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La operación de conjugar puede ser interpretada como una reflexión sobre el eje real, -b) está asociado con (a - ib) ib) es la reflexión a debido a que el punto par ordenado (a, -b) -b) asociado con (a + b). b). través del eje horizontal en el punto (a, -b) asociado
y
. (a, b) ( a+bi)
x
. (a, b)
( a-bi)
TEOREMA 1
Sea z y w números complejos entonces se tiene los siguientes enunciados: TEOREMA 1 1. 2. 3. 4. 5. 6. 7. 8. EJEMPLO:
̅ ̅ || |̅|̅ |||||| ̅ ||̅ || ||
̅
id demostrar que I zw I = I z I I w I 1) Sean dos números complejo z = a + bi y w = c + id demostrar
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DIVISION COMPLEJA Suponga que se quiere formar un cociente z/w donde z/w donde w es diferente de cero, este cociente es el numero complejo u tal que w *u = z. z. Sin embargo, esto nos ayuda para encontrar el valor de u. EJEMPLO:
DESIGUALDADES Hay varias desigualdades que tendrán ocasión de usarse.
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ARGUMENTO Y FORMA POLAR DE NÚMERO COMPLEJO Sea z = a + bi un bi un numero complejo distinto de cero. Entonces el par ordenado (a, b) es b) es un punto distinto del origen en el plano. Este punto tiene por coordenadas polares (r, θ), θ), como es notorio en las coordenadas polares el ángulo polar θ del par ordenado (a, b), b), no está únicamente determinado. Si camina en el eje real r unidades unidades a la derecha desde el origen y rota este segmento θo radianes desde el extremo del segmento en el par ordenado (a, b) como se muestra en la figura:
y (a, b)
x b) es cualquier número: θo + 2nπ, 2nπ, en el cual n es El ángulo polar θ para el punto (a, b) cualquier entero. Una elección positiva corresponde el giro desde cero hasta r , un radian b) en el sentido contrario al movimiento de las manecillas inicial θo para alcanzar al punto (a, b) en del reloj. El sentido negativo será en el movimiento de las manecillas del reloj: EJEMPLO:
√
√
(1,1) se puede denotar en forma polar ( Este tiene el punto (1,1) se polares del punto (1,1), (1,1), tendrá la forma ( , π/4 + 2nπ,). 2nπ, ).
, π/4), π/4), todas las coordenadas
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Según la fórmula del Euler se tiene o . Si θ bi, entonces el punto (a, b) tiene coordenadas polares par es cualquier argumento de z = a + bi, θ). ordenado (r, θ). y (a, b) r
|| )
x
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EJEMPLO: 1) Encontrar la forma polar de z = -1 + 4i y
(-1,4)
√ √
x
Debido a las propiedades exponenciales algunos cálculos con números complejos se simplifican si se usan las formas polares digamos que se quiere probar I zw I = I z I I w I
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Multiplicar dos números complejos tiene el efecto de sumar sus argumentos, modulo o un múltiplo entero de .
EJEMPLO: 1) Realizar la operación zw zw,, sabiendo que:
Img i (2, 2i)
2
-2
(2, -2i)
Real
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Lugares geométricos y conjuntos de pruebas Algunas veces la notación compleja es muy eficiente en la especificación especificación de lugares geométricos de puntos en el plano, la representación compleja de ciertos conjuntos que aparecen frecuentemente en las aplicaciones e integrales complejas.
DISTANCIA
||
Es cualquier numero compleja, el módulo de va hacer igual
| | | ||
la distancia del origen al número complejo
en el plano complejo. Si
es es
también es un
es la distancia entre en el plano complejo. Esta es la forma estándar de geometría para la distancia entre los puntos
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y
| | r
a
x
| | |
Esta es la manera de especificar a los círculos, en el plano complejo y muchas veces se hace referencia .
|| ||
Si entonces cualquier punto en el circulo tiene la forma polar es el ángulo desde la parte positiva del eje real a la recta desde el origen hasta .
donde
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| | | |
Esta expresión requiere que este en la bisectriz perpendicular del segmento de la recta que conecta a , la ecuación debe ser considerada como la ecuación de la recta. Ejemplos Encontrar la ecuación de la recta de la siguiente expresión.
|| | || | || ̅ | | || | ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅
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y=-13/9
i) ii) iii) iv)
||| | | | | | | | || | | | || | | |
Ejemplo
Indicar la región que representa las siguientes expresiones i)
ii)
| | ||
r=1
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vi)
120
FUNCIONES DE VARIABLE COMPLEJA
Una función de una variable compleja es una aplicación de un dominio de un subconjunto de los complejos . Donde la variable le aplicamos una función y obtenemos . A calos valor valor de le corresponde un valor de
; si a un valor de le corresponde más de un
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FUNCIÓN EXPONENCIAL
i)
, -
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,
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Sea
. Si =
FUNCIÓN LOGARÍTMICA
entonces al sacar el
;
Algunos conceptos conceptos importantes que debemos debemos conocer, en lo que tiene que ver a los lugares geométricos, siendo elementos de los números complejos.
esto representa el modulo y es la distancia desde el origen al punto
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Sea S un conjunto de números complejos y sea z en S entonces z es un punto de frontera de S o un punto inferior de S. Sea S un conjunto de números complejos si S es abierto, entonces S no puede contener ningún punto de frontera.
CONJUNTO CERRADO Un conjunto de números complejos es cerrado si contiene todos sus puntos frontera.
PUNTO LÍMITE
Un número es un punto límite de S si hay puntos de S arbitrariamente cercano a diferentes en . Un número complejo
pero
es un punto límite en un conjunto S di todo disco abierto alrededor
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*+ | |
La sucesión compleja converge a un numero sí, todo cualquier numero numero N tal que se cumple .
existe un
Esto significa que puede hacer cada término tan cerca como quiero de L eligiendo n al menos tan grande como algún termino N, puesto de otro menara dada cualquier disco abierto alrededor de puede encontrar algún termino de la sucesión de manera que todos los términos de la lista a partir de L caiga en D. esta es la idea más clara que está detrás de la convergencia de sucesiones reales excepto de que los intervalos abiertos en la recta real son reemplazados por discos abiertos. Cuando la sucesión converge a L se describe t o el límite de la sucesión compleja.
Determine las siguientes expresiones
*+
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SUSECION ACOTADA
Una sucesión se dice acotada si está acotada superior e inferiormente. Es decir si hay un número k menor o igual que todos los términos de la sucesión y otro K' mayor o igual que todos los términos de la sucesión. Por lo que todos los términos de la sucesión están
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LIMITES
Sea la función definida unívoca en una vecindad de con la posible excepción de Decimos que le número de es el límite de cunado tiende a y escribimos que
.
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LÍMITES AL INFINITO Por medio de la transformación de punto
osea el origen se aplica
se llama punto
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2. 3.
* +
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