Complex analysis: U-V transformations.

[Lavabug]

I'm a bit lost on this part of my course (ODE's and complex analysis). We've only done about 2-3 of these (seemingly simple) problems where we're given the equation of a line or circle in the complex plane and are asked to find its image in the U-V plane with some transformation $$\omega$$, but I really don't know what I'm supposed to do. I've checked Spiegel's complex variables book but I can't get a feel for what I'm doing. Can anyone please explain this to me and/or direct me to some examples?

 

[kru_]

Say you're given some point in the complex plane, such as 3 + 4i. If you then apply some function to that point, such as f(z) = 4z, you have a transformation on that point. In this case, f(3+4i) = 4(3 + 4i) = 12 + 16i. The original point 3+4i is mapped to the new point 12+16i.

An equation is just a representation of a bunch of different points. Each of those points is going to have a corresponding point after the transformation has been applied.

For example, the equation |z| = 1 represents a circle of radius 1, centered at the origin. The above transformation applied to every point in |z| = 1 will result in a new circle that is 4 times bigger, but still centered at the origin.