Page 32 - Calculus with Complex Numbers

Prove that for all IzI = R > l

1 1
:% z .
22 + z + 1 R - R - 1

Prove that Iez I= clke:

Find where I ez Iis maximum for Iz Is l (draw a diagram).

Prove that for z = .x + iy
I sint.v + iy) 12 = sinz.x + sinhzy,
I cost.x + iy) 12 = cosz.x + sinhzy.

Find where I sin z Iis maximum for I z Is 1 (draw a diagram).

Prove that all points z satisfying
z+ 1
= 2
z+ 4
lie on a circle. Find its centre and radius.