*Funcin ExponencialSea z = x+iy, definimos la funcin exponencial como:

*Funcin ExponencialPropiedades

(1) ez se reduce a ex cuando z es real (2) ez es una funcin entera. (3) (ez )=(ez )

*

*(2) Frmula de Euler(3) Las formas exponencial y trigonomtrica

*Aplicacin: FasoresMuchas seales pueden ser representadas como senoides:

*Representacin de un nmero complejo en forma de fasor

*Corriente AlternaCircuitos

*Podemos escribir:Funciones trigonomtricasA partir de la frmula de Euler:

*Funciones trigonomtricas de variable compleja

*Funciones trigonomtricas de variable complejaPropiedades:cos z (sin z) se reduce a cos x (sin x) cuando z es real.(2) cos z y sin z son funciones enteras(3) Sus derivadas coinciden con sus equivalentes en variable real.

*Resolver cos z = 1.OTRAS FUNCIONES TRIGONOMETRICAS:

*Propiedadestan z y sec z (cot z y csc z) no son enterasSe cumple:

*Funciones hiperblicas complejas

*Funciones hiperblicas complejas(1) Estas funciones son enteras(2) Otras funciones hiperblicas se definen como:tanh z = sinh z / cosh z ; coth z = cosh z / sinh zsech = 1/cosh z ; csech z = 1/sinh zPropiedades(3) Se cumple:

*Funcin logartmicaSe Define el logaritmo de un nmero complejo z como(|z| > 0)

*Funcin logartmicaPropiedadesSe cumple:El logaritmo complejo es multivaluado(ln z)/ = 1/z

*PotenciasDefinamos ahora donde c = a+bi es complejo como:

*Si c = n = .-1,-2,1,2,.... entonces zn es univaluadoSi c = p/q, siendo el cociente de dos enteros positivos, zc tiene un nmero finito de valores distintos.Si c es irracional o complejo entonces zc es infinitamente multivaluado.Propiedades:Ejemplo: Calcular ii

****1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*****1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.**1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.*1. The real variable can be represented by a single line. All the values it can take, from -inf to +inf can be plotted on this line2. We usually denote the funcin of x by the letter y - each point along the x-axis es mapped to a point in the x-y plane, y we can visualise the funcin of the real variable.**