For the transformation w=z^2, find the locus of w when z lies on the imaginary axis.
I thought the locus would be a line along the real axis as I thought w= r^2 or -r^2 but the book says it has argument pi which would mean only on the negative real axis ie w= -r^2 why both are not valid where have I gone wrong
To find the locus of points when z lies on the imaginary axis and the transformation w = z^2 is applied, we can start by considering a complex number z on the imaginary axis, which can be written as z = iy, where y is a real number and i is the imaginary unit.
Now, applying the transformation w = z^2, we substitute z = iy into the equation:
w = (iy)^2
w = -y^2
So, the transformed complex number w will be on the negative real axis, represented by -y^2. Here, it's important to note that the locus is not just a line along the real axis, but specifically the negative real axis.
In polar form, a complex number can be expressed as z = r * exp(iθ), where r represents the magnitude of the complex number and θ represents its argument.
In this case, we have w = -y^2, where y is the magnitude of the complex number. However, we don't have any information about the argument θ. This means that the locus of w will have all possible arguments, which includes both π (pi) and -π (negative pi).
Therefore, the correct locus for the given transformation w = z^2, when z lies on the imaginary axis, is on the negative real axis (w = -y^2) with arguments π and -π.