Linear Transformation of Complex Analysis Problems

Question

What is the result of linear transformation of the circle $L: |z-1|=1$ by $w = \dfrac{i} {z+2i}$?

I don't have idea how to step up first on this problem. I have read some texts, but don't found any informations to solve this problems. Do you have any idea? At least giving me step to do.

Answer 1

Step 1: Invert the transformation, i.e., find $z$ in terms of $w$. Hint: $\pmatrix{0 & i \\ 1 & 2i} \pmatrix{-2i & i \\ 1 & 0} = \pmatrix{i & 0 \\ 0 & i}$.

Step 2: Put this expression for $z$ into $|z-1| = 1$ to get an equation in $w$.

Step 3: Look at this $w$-equation to see what kind of curve it describes.

Answer 2

Rearrange $w=\dfrac{i}{z+2i}$ to get $z=\dfrac{i}{w}-2i$. Substitute into $L$:

$$|\frac{i}{w}-2i-1|=1 \iff |i-(2i+1)w|=|w|$$ Now let $w=x+iy$ and square both sides: this should help you to work out what the transformation does.