Complex Analysis homework
Describe geometrically the sets of points z in the complex plane defined by the following relations:
(1) |z − z1| = |z − z2| where z1, z2 ∈ C.
(2) 1/z = z.
(3) Re(z) = 3.
(4) Re(z) > c, (resp., ≥ c) where c ∈ R.
(5) Re(az + b) > 0 where a, b ∈ C.
(6) |z| = Re(z) + 1. (7) Im(z) = c with c ∈ R.
Let h·, ·i denote the usual inner product in R 2 . In other words, if Z = (z1, y1) and W = (x2, y2), then hZ, Wi = x1x2 + y1y2. Similarly, we may define a Hermitian inner product (·, ·) in C by (z, w) = zw. The term Hermitian is used to describe the fact that (·, ·) is not symmetric, but rather satisfies the relation (z, w) = (w, z) for all z, w ∈ C. Show that hz, wi = 1 2 [(z, w) + (w, z)] = Re(z, w), where we use the usual identification z = x + iy ∈ C with (x, y) ∈ R 2 .
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