For the transformation w=z^2, show that as z moves once round the circle centre O and radius 2, w moves twice round centre O radius 4.
I am okay finding the radius and centre of the locus but how do I show it moves twice roundfor w for z turning once.
See my other post. Transform to polar coordinates, you get a double angle in the transformation.
Thanks
To show that as z moves once around the circle centered at O with radius 2, w moves twice around the circle centered at O with radius 4, we can use the transformation w = z^2 and analyze it in polar coordinates.
Let's represent z and w in polar form:
z = r1 * e^(iθ1)
w = r2 * e^(iθ2)
Here, r1 and r2 represent the magnitudes of z and w, respectively, and θ1 and θ2 represent the angles they make with the positive real axis.
Now, substitute these polar representations into the transformation equation w = z^2:
r2 * e^(iθ2) = (r1 * e^(iθ1))^2
r2 * e^(iθ2) = r1^2 * e^(2iθ1)
Comparing the magnitudes:
|r2 * e^(iθ2)| = |r1^2 * e^(2iθ1)|
|r2| = |r1^2|
Since r1 is the radius of the starting circle (radius 2) and r2 is the radius of the resulting circle (radius 4), we can see that |r2| = |r1|^2.
This means that the radius r2 is equal to the square of the radius r1. In other words, as z moves once around the circle centered at O with radius 2, the corresponding w moves along a circle centered at O with radius 4.
However, we need to show that w moves twice around the circle centered at O with radius 4.
To do this, let's consider the angle θ2 in the transformation equation. From the equation, we have:
e^(iθ2) = e^(2iθ1)
The exponential function e^(ix) represents rotation in the complex plane. When we multiply the exponent by 2, it corresponds to doubling the angle.
Since θ2 is double the angle θ1, as z moves once around the circle, θ1 completes a full revolution (360 degrees or 2π radians), and θ2 completes two full revolutions (720 degrees or 4π radians).
Therefore, as z moves once around the circle centered at O with radius 2, w moves twice around the circle centered at O with radius 4.