Find the lmage of.
under the mapping

To start, the equation represents a circle in the complex plane centered at with a radius of 2. The center can be represented as the point (0, 2) in the Cartesian plane. The circle will include all complex numbers such that the distance from to the point is exactly 2.

When we consider the mapping , we need to express in terms of the circle’s parametric representation. We can write as:

where indicates:

Squaring both sides gives:

To find the image under the mapping , it’s helpful to parameterize the circle. The points on the circle can be expressed as:

for ranging from to . This gives:

Thus, the image of the circle under the mapping requires calculating :

We can further analyze the expression for :

To simplify this complex fraction, multiply the numerator and the denominator by the conjugate of the denominator:

The denominator simplifies to .

Now, since is a circle centered at with radius 2, the image of this circle under the mapping will be a new curve in the complex plane.

This transformation generally converts circles into arcs or circles, depending on the radius and position relative to the origin. Specifically, since the original circle does not include the origin, its image under the transformation will also avoid the origin but will create a more complex contour that can be visualized further by plotting or using computational software.

In summary, the image of the circle under the mapping results in a new curve, which could be explored more specifically by considering various values of and tracing the resulting points in the complex plane.