Question

Find the image of y=3x under w=1/z

Answer 1

Answer:

Under the transformation w=z2, find the images of

straight line y=x

"rayo in spanish" argz=θ

I try

1-

K={z=x+iy:y=x}

w=(x+iy)2=x2−y2+2ixy=2y2

then u=0 and v=2y2

f(K)={w=u+iv:v=2y2,y∈R}You have already in the first case that

y=x⟹u=0,v=2x2

so the locus is the positive imaginary axis.

In the second case,

argw=argz2=2argz=2θ

so the locus is the part-line

argw=2θ