A First Course in Complex Analysis · Problem 53

Integration — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-functions-geometry

Problem

As in Exercise 3.6.52, let

f(z) = z^2

. Let

Q

be the square with vertices at

0, 2, 2+2i

and

2i

. Draw

f(Q)

and identify the types of image curves corresponding to the segments from

2

2+2i

and from

2+2i

2i

. They are not parts of either straight lines or circles. Hint. You can write the vertical segment parametrically as

z(t) = 2 + it

. Eliminate the parameter in

u + iv = f(z(t))

to get a

(u,v)

equation for the image curve.