Variable Compleja
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Dada la función de variable compleja f(z) comprobar si es analítica. En caso afirmativo determine su derivada
z=x + iy f(z) = cos(3z) w = cos(3(x + iy)) w = cos(3x + 3iy)) cos z = e^(zi)+e^(-zi)/2 COS Z = ½(e^(3x +3iy)i +e^-(3x +3iy)i) W = ½(e^(-3y) e^(-3xi) + e^(3y) e^(3xi) ) e^(iθ) = cos θ + sen iθ w = ½ { e^(-3y) { cos (3x) + isen(3x)} + e^(3y){ cos(-3x) + isen(-3x)} w = cos (3x)/2 (e^(3y) + e^(-3y) ) + sen (3x)/2 (e^(3y) e^(-3y) ) u = cos (3x)/2 (e^(3y) + e^(-3y) ) v = sen (3x)/2 (e^(-3y) - e^(3y) ) du/dx ( 1) dv/dy ( 2)
=
{ =
-1/2
sen3x
(e^(3y)
+
(-3sen(3x)/2)*(e^(3y)
e^(-3y) +
)
–
3}
e^(-3y)
)
du/dy = (cos 3x/2) * (e^(3y) + e^(-3y) ) du/dy =3/2 cos 3x (e^(3y) + e^(-3y) ) du/dx = (cos 3x(3)/2)* (e^(3y) + e^(-3y) ) du/dy =-( dv/dy) 3/2 cos 3x (e^(3y) - e^(-3y) ) = -3/2 cos 3x (e^(3y) - e^(3y) )
3/2 cos 3x (e^(3y) - e^(-3y) ) =3/2 cos 3x (e^(3y) - e^(3y) ) f(z) es analítica df (cos 3z)=- 3/2 sen 3x (e^(3y) + e^(-3y) ) + i3/2 sen 3x (e^(-3y) - e^(3y) )
z=x + iy f(z) = cos(3z) w = cos(3(x + iy)) w = cos(3x + 3iy)) cos z = e^(zi)+e^(-zi)/2 COS Z = ½(e^(3x +3iy)i +e^-(3x +3iy)i) W = ½(e^(-3y) e^(-3xi) + e^(3y) e^(3xi) ) e^(iθ) = cos θ + sen iθ w = ½ { e^(-3y) { cos (3x) + isen(3x)} + e^(3y){ cos(-3x) + isen(-3x)} w = cos (3x)/2 (e^(3y) + e^(-3y) ) + sen (3x)/2 (e^(3y) e^(-3y) ) u = cos (3x)/2 (e^(3y) + e^(-3y) ) v = sen (3x)/2 (e^(-3y) - e^(3y) ) du/dx ( 1) dv/dy ( 2)
=
{ =
-1/2
sen3x
(e^(3y)
+
(-3sen(3x)/2)*(e^(3y)
e^(-3y) +
)
–
3}
e^(-3y)
)
du/dy = (cos 3x/2) * (e^(3y) + e^(-3y) ) du/dy =3/2 cos 3x (e^(3y) + e^(-3y) ) du/dx = (cos 3x(3)/2)* (e^(3y) + e^(-3y) ) du/dy =-( dv/dy) 3/2 cos 3x (e^(3y) - e^(-3y) ) = -3/2 cos 3x (e^(3y) - e^(3y) )
3/2 cos 3x (e^(3y) - e^(-3y) ) =3/2 cos 3x (e^(3y) - e^(3y) ) f(z) es analítica df (cos 3z)=- 3/2 sen 3x (e^(3y) + e^(-3y) ) + i3/2 sen 3x (e^(-3y) - e^(3y) )
