Determine the image of the circle S = {z ∈ C | |z−1|= 1} by the function f(z) = z2−2z. The equation of the circle S is given by α(t) = 1 + cos(t) ...
Determine the image of the circle S = {z ∈ C | |z−1|= 1} by the function f(z) = z2−2z.
The equation of the circle S is given by α(t) = 1 + cos(t) + isin(t), t ∈ [0, 2π].
The equation of the function f(z) is given by f(z) = (z-1)^2 - 1.
The roots of the equation f(z) = α are solutions in Dom(f) that make f(β) = α.
The image of S is a circle with center (-1, 0) and radius 1.
a) I, II and IV are correct.
b) II and III are correct.
c) III and IV are correct.
d) I and III are correct.
The equation of the circle S is given by α(t) = 1 + cos(t) + isin(t), t ∈ [0, 2π].
The equation of the function f(z) is given by f(z) = (z-1)^2 - 1.
The roots of the equation f(z) = α are solutions in Dom(f) that make f(β) = α.
The image of S is a circle with center (-1, 0) and radius 1.
a) I, II and IV are correct.
b) II and III are correct.
c) III and IV are correct.
d) I and III are correct.
💡 1 Resposta
Ed 
Para determinar a imagem do círculo S = {z ∈ C | |z−1|= 1} pela função f(z) = z² - 2z, podemos substituir z por α(t) = 1 + cos(t) + i sin(t), onde t ∈ [0, 2π], que é a equação do círculo S. Então, temos: f(α(t)) = (α(t) - 1)² - 2(α(t)) = (cos(t))^2 + i(2sin(t) - sin(t)^2 - 2cos(t)) Assim, a imagem do círculo S pela função f(z) é um novo círculo com centro (-1, 0) e raio 1. Portanto, a alternativa correta é a letra d) I e III estão corretas.
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