A First Course in Complex Analysis · Problem 13

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · mobius-transformations

Problem

Let

f(z) = \frac{2z}{z+2}

. Draw two graphs, one showing the following six sets in the

z

-plane and the other showing their images in the

w

-plane. Label the sets. (You should only need to calculate the images of

0, \pm 2, \pm(1+i)

, and

\infty

; remember that Möbius transformations preserve angles.) ((a)) the

x

-axis plus

\infty

((b)) the

y

-axis plus

\infty

((c)) the line

x = y

plus

\infty

((d)) the circle with radius 2 centered at 0 ((e)) the circle with radius 1 centered at 1 ((f)) the circle with radius 1 centered at

-1