Variable compleja
Empezamos con un nuevo tema: Variable compleja. Del semestre pasado tengo algunos ejercísios para resolver, aquí en clase:
Problema T05.3 (b,c) @Sahid
Hola Sahid, podrías poner to solución de estos incisos en el blog. Y explicar un poco se es necesario para que los demas lo entiendan
a) z'=z_1*z_2
ResponderBorrarz'=(x_1 + i y_1)*(x_2+iy_2)
z'=x_1*x_2 +i(x_1*y_2) +i(x_2*y_1) + i^2*y_1*y_2
z'=(x_1*x_2 - y_1*y_2) + i(x_1*y_2 +x_2*y_1)
Bien!
Borrarb) z'= z_1/z_2
ResponderBorrarz'= x_1+iy_1/(x_2+iy_2)
z'= (x_1+iy_1)/(x_2+iy_2)*[(x_2 -iy_2)/(x_2-iy_2)]
z'= (x_1*x_2 +y_1*y_2 + i(x_2*y_1 - x_1*y_2))/(x_2^2 + y_2^2)
z'=[(x_1*x_2 + y_1*y_2)/(x_2^2 + y_2^2)] +i[(x_2*y_1 - x_1*y_2)/(x_2^2 + y_2^2)]
Bien!
Borrarc) |z|=sqrt(x^2+y^2) ; \theta= arctan(y/x)
ResponderBorrarz_1=1 + i
|z_1|= sqrt(1+1)= sqrt(2)
\theta_1= arctan(1)= pi/4
z_1= sqrt(2)*exp(i*pi/4)= sqrt(2)*(cos(pi/4) + isen(pi/4))
z_2= -1 + i
|z_2|=sqrt(2)
\theta_2 = pi - arctan(1) = pi - pi/4 = 3pi/4
z_2= sqrt(2)*exp(i*3pi/4)= sqrt(2)*(cos(3pi/4) +isen(3pi/4))
z_3= - 1 - i
|z_3|=sqrt(2)
\theta_3= pi + arctan(1)= pi + pi/4= 5pi/4
z_3=sqrt(2)*exp(i*5pi/4)=sqrt(2)*(cos(5pi/4) + isen(5pi/4))
z_4= 1 + isqrt(3)
|z_4|= sqrt(4)= 2
\theta_4= arctan(sqrt(3)) = pi/3
z_4= 2*exp(i*pi/3)=2*(cos(pi/3) + isen(pi/3))
z_5= -1 -isqrt(3)
|z_5|=2
\theta_5= pi + arctan(sqrt(3))= pi + pi/3 = 4pi/3
z_5= 2*exp(i*4pi/3)=2*(cos(4pi/3) + isen(4pi/3))
z_6= sqrt(3) - i
|z_6|=2
\theta_6= 2pi- arctan(1/sqrt(3))= 2pi - pi/6= 11pi/6
z_6= 2*exp(i*11pi/6)= 2*(cos(11pi/6) + isen(pi/6))
z_7= 2 + sqrt(3) + i
|z_7|=sqrt( (2+sqrt(3))^2 +1)= sqrt(4+3+4*sqrt(3) +1)=sqrt(8 +4*sqrt(3))= 2*sqrt(2+sqrt(3))
\theta_7= arctan(1/(2 + sqrt(3)))= arctan([1/(2+sqrt(3)]*[(2-sqrt(3))/(2-sqrt(3))])=arctan(2-sqrt(3))
z_7=2*sqrt(2+sqrt(3))*exp(i*arctan(2-sqrt(3))=2*sqrt(2+sqrt(3))*(cos(arctan(2-sqrt(3)) + isen(arctan(2-sqrt(3))
Caracoles
BorrarMadre de cristo no vi que esto era de tarea!
BorrarAquí van:
A) z' = z_1*z_2 = (x_1+iy_1)(x_2+iy_2) = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)
B) z' = z_1 / z_2 = (x_1+iy_1) / (x_2+iy_2) = (x_1+iy_1) / [(x_2+iy_2) * (x_2-iy_2)] =
[(x_1+iy_1) * (x_2-iy_2)] / (x_2 ^2 + y_2 ^2) =
[x_1x_2+ y_1y_2 + i(x_2y_1 - x_1y_2)] / (x_2 ^2 + y_2 ^2)
C)
1) |z| = sqrt(1^2+1^2) = sqrt(2), phi = tan^-1 (1) = pi/4 -> sqrt(2) e^i(pi/4)
2) |z| = sqrt((-1)^2+1^2) = sqrt(2), phi = pi - tan^-1 (1) = 3pi/4 -> sqrt(2) e^i(3pi/4)
3) |z| = sqrt((-1)^2+(-1)^2) = sqrt(2), phi = pi + tan^-1 (1) = 5pi/4 -> sqrt(2) e^i(5pi/4)
4) |z| = sqrt(1^2+sqrt(3)^2) = 2, phi = tan^-1 (sqrt (3)) = pi/3 -> 2*e^i(pi/3)
5) |z| = sqrt((-1)^2+(-sqrt(3))^2) = 2, phi = pi + tan^-1 (sqrt (3)) = 4pi/3 -> 2*e^i(4pi/3)
6) |z| = sqrt((-1)^2+sqrt(3)^2) = 2, phi = 2pi - tan^-1 (1/sqrt (3)) = 2pi - pi/6 = 11pi/6
-> 2*e^i(11pi/6)
7) |z| = sqrt(1^2+sqrt(2+sqrt(3)) ^2) = 2sqrt (2+sqrt(3)),
phi = tan^-1 (1/ sqrt(2+sqrt(3))) = 5pi/12 -> 2sqrt (2+sqrt(3))*e^i(5pi/12)
a)
ResponderBorrarz_1 z_2 = (x_1 + iy_1)(x_2 + iy_2)
z_1 z_2 = (x_1 x_2 - y_1 y_2) + i(x_1 y_2 + x_2 y_1)
b)
\frac{z_1}{z_2} = \frac{(x_1 + iy_1)}{(x_2 + iy_2)}
\frac{z_1}{z_2} = \frac{(x_1 + iy_1)(x_2 - iy_2)} {(x_2 + iy_2)(x_2 - iy_2)}
\frac{z_1}{z_2} = \frac{(x_1 x_2 + y_1 y_2) + i(x_1 y_2 + x_2 y_2)} {(x_2^2 + y_2^2)}
c)
1. \sqrt{2} e^{i\frac{\pi}{4}}
2. \sqrt{2} e^{i\frac{3 \pi}{4}}
3. \sqrt{2} e^{i\frac{5 \pi}{4}}
4. 2 e^{i \arctan{\sqrt{3}}}
5. 2 e^{i (\pi + \arctan{\sqrt{3}} ) }
6. 2 e^{i (\frac{3 \pi}{2} + \arctan{\sqrt{3}} ) }
7. \sqrt{8+4\sqrt{3}} e^{i \arctan{\sqrt{2 + \sqrt{3}}}}
a) z' = z_1 * z_2
ResponderBorrar= (x_1 + i y_1) * (x_2 + i y_2)
= (x_1*x_2 - y_1*y_2) + i ( x_1*y_2 + x_2*y_1)
b) z' = z_1/z_2
= ((x_1 + i y_1) / (x_2 + i y_2)) * (x_2 - i y_2) / (x_2 - i y_2)
= (x_1 * x_2 + y_1 * y_2)/(x_2^2 + y_2^2) + i (y_1 * x_2 - x_1 * y_2)/(x_2^2 + y_2^2)
c)
1) 1 + i = sqrt(2) * exp(i * pi/4)
2) -1 + i = sqrt(2) * exp(i *3pi/4 )
3) -1- i = sqrt(2)* exp(i * 5pi/4)
4) 1 + i*sqrt(3) = 2 * exp(i * pi/3)
5) -1 - i*sqrt(3) = 2 * exp(i * 4pi/3)
6) sqrt(3) - i = 2 * exp(i * 11pi/6)
7) 2 + sqrt(3) + i = 2 * sqrt (2 + sqrt(3)) * exp(i * pi/6)
PUBLICACIÓN ATRASADA:
ResponderBorrar(a)
z'=z1*z2=(x1+iy1)*(x2+iy2)=x1*x2+ix1*y2+ix2*y1-y1*y2
=(x1x2-y1y2)+i(x1y2+x2y1)
(b)
z'=(x1+iy1)/(x2+iy2)=((x1+iy1)/(x2+iy2))*((x2-iy2)/(x2-iy2))=(x1x2-ix1y2+ix2y1+y1y2)/(x2^2-ix2y2+ix2y2+y2^2)
=((x1x2+y1y2)/(x2^2+y2^2))+i((x2y1-x1y2)/(x2^2+y2^2))
(c)
(1) phi=pi/4, |z|=sqrt(2)
z=sqrt(2)*exp(ipi/4)=sqrt(2)*(cos(pi/4)+isin(pi/4))
(2) phi=3*pi/4, |z|=sqrt(2)
z=sqrt(2)*exp(i3pi/4)=sqrt(2)*(cos(3*pi/4)+isin(3*pi/4)
(3) phi=-3*pi/4, |z|=sqrt(2)
z=sqrt(2)*exp(-i3pi/4)=sqrt(2)*(cos(3pi/4)+isin(3pi/4))
(4) phi=pi/3, |z|=2
z=2*exp(ipi/3)=2*(cos(pi/3)+isin(pi/3)
(5) phi=-2pi/3, |z|=2
z=2*exp(-i2pi/3)=2*(cos(2pi/3)-isin(2pi/3)
(6) phi=-pi/6, |z|=2
z=2*exp(-ipi/6)=2*(cos(pi/6)-isin(pi/6)
(7) phi=pi/12, |z|=2*sqrt(2+sqrt(3))
z=2*sqrt(2+sqrt(3))*exp(ipi/12)=2*sqrt(2+sqrt(3))*(cos(pi/12)+isin(pi/12))
a)
ResponderBorrar$z' = z_1 * z_2$\\
$= (x_1 + i y_1)*(x_2 + i y_2)$\\
$= (x_1*x_2 - y_1*y_2) + i ( x_1*y_2 + x_2*y_1)$
b)
$z' = z_1/z_2$\\
$= ((x_1 + i y_1) / (x_2 + i y_2)) * (x_2 - i y_2) / (x_2 - i y_2)$\\
$= (x_1 * x_2 + y_1 * y_2)/(x_2^2 + y_2^2) + i (y_1 * x_2 - x_1 * y_2)/(x_2^2 + y_2^2)$\\
c)
$z_1 = \sqrt{2} * \exp^{(i* \pi/4)}$\\
$z_2 = \sqrt{2} * \exp^{(i *3 \pi/4 )}$\\
$z_3 = \sqrt{2}* \exp^{(i * 5 \pi/4)}$\\
$z_4 = 2 * \exp^{(i * \pi/3)}$\\
$z_5= 2 * \exp^{(i * 4\pi/3)}$\\
$z_6 = 2 * \exp^{(i * 11\pi/6)}$\\
$z_7 =2* \sqrt{(2+\sqrt{3})} * \exp^{(i*\arctan {(2-\sqrt{3})}}$\\
a) z' = z_{1} * z_{2} = (x_{1} + iy_{1}) * (x_{2} + iy_{2})
ResponderBorrar= x_{1} * x_{2} + i( x_{1} * y_{2} + x_{2} * y_{1}) - y_{1} * y_{2}.
b) z' = \frac{ z_{1} }{ z_{2} } = \frac{ x_{1} + iy_{1} }{ x_{2} + iy_{2} }
= (\frac{ x_{1} + iy_{1} }{ x_{2} + iy_{2} } ) * ( \frac{ x_{2} - iy_{2} }{ x_{2} - iy_{2} } )
= \ frac{ x_{1} * x_{2} + y_{1} * y_{2} }{ x_{2}^{2} + y_{2}^{2} }
+ i * \frac{ x_{2} * y_{1} - x_{1} * y_{2} }{ x_{2}^{2} + y_{2}^{2} }
c) En forma polar: , si \theta = arctan \frac{y}{x}
(1) => \sqrt{2} * e^i{\pi/4} = \sqrt{2} * ( \cos{\pi/4} + i * \sin{\pi/4} )
(2) => \sqrt{2} * e^i{3 * \pi/4} = \sqrt{2} * ( \cos{3*\pi/4} + i * \sin{3*\pi/4} )
(3) => \sqrt{2} * e^i{5 * \pi/4} = \sqrt{2} * ( \cos{5*\pi/4} + i * \sin{5*\pi/4} )
(4) => 2 * e^i{\pi/3} = 2 * ( \cos{\pi/3} + i * \sin{\pi/3} )
(5) => 2 * e^i{4 * \pi/3} = 2 * ( \cos{4 * \pi/3} + i * \sin{4 * \pi/3} )
(6) => 2 * e^i{11* \pi/6} = 2 * ( \cos{11 * \pi/6} + i * \sin{11* \pi/6} )
(7) => 2 * \sqrt{ 2 + \sqrt{3} } * e^i{\pi/12} = 2 * \sqrt{ 2 + \sqrt{3} } * ( \cos{ \pi/12} + i * \sin{ \pi/12} )