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I NUMERI COMPLESSI

matematica


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I numeri complessi





J = unitā immaginaria

J =√ -1

J2 = -1



Piano di gauss (complesso;argand-gauss)






                     242h76c           I



                     242h76c     y Z(x ;y)




                     242h76c     0 x R

                     242h76c                      242h76c                   


X+JY forma algebrica (forma cartesiana)


X = reale

Y = immaginaria          


Es:                     242h76c                      242h76c                      242h76c                      242h76c          I numeri senza parte reale sono                      242h76c                      242h76c                      242h76c                      242h76c detti immaginari puri

                     242h76c      Z Re Imm


                     242h76c     6 6 0

-3+j4 -3 4

-2-j5 -2 -5                     242h76c               I lungh vettore: Z

                     242h76c     j4 0 4                     242h76c                      242h76c        angolaz: φ



                     242h76c                      242h76c                      242h76c                      242h76c                      242h76c                   Z


SOMMA:

Zs: Z1+Z2 = (X1+X2)+j(J1+J2)                     242h76c                      242h76c     φ

                     242h76c                      242h76c                      242h76c                      242h76c                      242h76c      

PRODOTTO:                     242h76c                      242h76c                      242h76c                                  242h76c                      242h76c            R

Zp:Z1*Z2


CONIUGAZIONE:                     242h76c                      242h76c                      242h76c                      242h76c        Z*

Z :x jy 5*(1-j)

Z*:x-jy 5*(1+j)


Complesso coniugato

MODULO:                     242h76c                      242h76c              r = modulo

r = √ x 2+y 2

                     242h76c                      242h76c                      242h76c           

                     242h76c                      242h76c                      242h76c            complesso coniugato di Z2

DIVISIONE:


Z1 x1+jy1 * x2 jy2

Z2        x2+jy2 x2 jy2



Forma polare o di Steinmetz

Z≡(R;


׀ I                      242h76c    ׀


                     242h76c                      242h76c                      242h76c

                     242h76c                      242h76c                      242h76c Z                     242h76c              Z=X+jY


                     242h76c                  

                     242h76c       r                     242h76c        y

                     242h76c                                        242h76c                      242h76c                 Z =r *   

0                     242h76c                R



                     242h76c                      242h76c                 V                   




= arctg * y                     242h76c                      242h76c

x                      242h76c                      242h76c                      242h76c            r= √ x 2+y 2




va bene se 1 e 4 quadrante                 

                     242h76c                      242h76c           con x≠0

                     242h76c                      242h76c                      242h76c

= arctg * y + 180°                     242h76c              

x

se 2 e 3 quadrante





x = 0 ;φ = +90 se vettore positivo

x = 0 ;φ = -90 se vettore negativo con x = 0









FORMULE INVERSE:


                     242h76c                   x

                     242h76c       i                     242h76c                      242h76c                      242h76c          x = r* cos φ

                     242h76c                      242h76c                      242h76c                      242h76c                  y = r* sin

                     242h76c                      242h76c                      242h76c y

                     242h76c       r   φ                     242h76c                   

                     242h76c                  

                     242h76c       0                     242h76c r



r = lungh vettore

φ = angolaz vettore  

                     242h76c                                        242h76c                      242h76c  



3) Forma trigonometrica


Z = r * φ


PRODOTTO:

Z1 * Z2 = r1*r2 * (φ1+ φ 2)


DIVISIONE:

Z1/Z2 = r1/r 2 * (φ 1- φ 2)


COMPLESSO CONIUGATO:


Z = r * φ                     242h76c    Z = r * -φ


4) : forma trigonometrica


● Z = r * (cos + j sen



PRODOTTO:

Z1*Z2 = r1*r2 *[cos(φ1+ φ 2)+jsen(φ 1+ φ 2)]


DIVISIONE:

Z1/Z2 = r1/r2*[cos*( φ 1- φ 2)+jsen(φ 1- φ 2)]


COMPLESSO CONIUGATO:


Z = r * (cos φ + j sen φ) Z = r * (cos φ - j sen φ)


5) Forma esponenziale


ez = exp


Z = r * ej


PRODOTTO:                     242h76c                      242h76c                      242h76c                      242h76c     nella f.esponenziale la φ č in

                     242h76c                      242h76c                      242h76c                      242h76c            radianti


Z1*Z2= (r1* ejφ1) *(r2* e)                     242h76c       r1*r2*e j(φ1+ φ2)                 



DIVISIONE


Z1/Z2= r1* ejφ1 =                      242h76c            r1    * e j(φ1-φ2)

r2* ejφ2                      242h76c           r2







                     242h76c                              242h76c                         242h76c            φ°= φ(rad)*π φ(rad)= φ°*180

                     242h76c                      242h76c                      242h76c                      242h76c        180                     242h76c                π




                     242h76c    


                     242h76c    

                     242h76c    












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