Loci — Regions and Inequalities

The recipe is the same in every case: draw the boundary (using the equality), choose solid vs dashed based on strictness, then test a single point to pick the side.

1. Disk. $|z - z_0| \leq r$ — closed disk centred at $z_0$ with radius $r$. Solid circle, shade the interior. $|z - z_0| < r$ — open disk (dashed circle).
2. Exterior of a disk. $|z - z_0| > r$ — shade everything outside the circle, with a dashed boundary.
3. Half-plane. $\mathrm{Re}(z) \geq a$ — vertical line $x = a$ (solid), shade $x \geq a$. $\mathrm{Im}(z) < b$ — horizontal line $y = b$ (dashed), shade $y < b$.

Worked through the hook: $\{z : |z| \leq 2 \text{ and } \mathrm{Re}(z) \geq 0\}$.

• $|z| \leq 2$: closed disk of radius 2 centred at the origin (solid boundary).
• $\mathrm{Re}(z) \geq 0$: closed half-plane $x \geq 0$ (solid boundary on the imaginary axis).
• Intersection: the right half of the closed disk — a closed semicircular region.

$|z - z_0| \leq r$: closed disk (solid boundary) · $|z - z_0| > r$: exterior of circle (dashed boundary) · $\mathrm{Re}(z) > a$: half-plane $x > a$ (dashed vertical line) · Always test the origin (or another easy point) to confirm which side to shade

Pause — copy the disk/exterior boundary conventions (solid for $\leq$, dashed for $>$, strict), the half-plane rules, and the test-a-point strategy into your book.

We just saw that $|z-z_0| \leq r$ is a closed disk (solid boundary) and $|z-z_0| > r$ the exterior (dashed boundary), with the test-a-point rule confirming which side to shade. That raises a question: how do argument inequalities determine a region? This card answers it → $\alpha < \arg(z) < \beta$ is a sector with apex at the origin; $\arg(z-z_0) = \theta$ translates the apex to $z_0$.

The argument $\arg(z)$ measures the angle from the positive real axis to the ray from $0$ to $z$, anticlockwise positive. An inequality $\alpha < \arg(z) < \beta$ shades the wedge of points whose argument lies between $\alpha$ and $\beta$:

• The two bounding rays $\arg(z) = \alpha$ and $\arg(z) = \beta$ start at the origin.
• Strict $<$: rays drawn dashed and the origin is excluded ($\arg(0)$ is undefined).
• Non-strict $\leq$: rays drawn solid (but the origin still has no argument, so technically still excluded).

Translating $\arg(z - z_0)$. Instead of an angle at the origin, $\arg(z - z_0)$ is the angle at $z_0$. So $\arg(z - z_0) = \theta$ is a ray starting at $z_0$ (not the origin), and an inequality between two such conditions gives a sector with apex at $z_0$.

$\arg(z) = \theta$ is a RAY from the origin, not a full line · $\alpha < \arg(z) < \beta$ is a sector with apex at the origin · $\arg(z - z_0) = \theta$ is a ray from $z_0$ — apex translates with the subtraction · The origin ($z = 0$) is always excluded from $\arg$ inequalities

Pause — copy $\arg(z)=\theta$ as a ray from the origin, $\alpha < \arg(z) < \beta$ as a sector, the translation rule for $\arg(z-z_0)$, and that $z=0$ is always excluded from argument conditions into your book.