A First Course in Complex Analysis · Problem 52

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-functions-geometry

Problem

For this problem,

f(z) = z^2

. ((a)) Show that the image under

f

of a circle centered at the origin is a circle centered at the origin. Hint. Use polar coordinates. ((b)) Show that the image under

f

of a ray starting at the origin is a ray starting at the origin. ((c)) Let

T

be the figure formed by the horizontal segment from

0

2

, the circular arc from

2

2i

, and then the vertical segment from

2i

0

. Draw

T

and

f(T)

. ((d)) Is the right angle at the origin in (c) preserved? Is something wrong here?