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Right Half-Plane — Definition, Formula & Examples

The right half-plane is the region of the coordinate plane (or complex plane) consisting of all points whose x-coordinate (or real part) is positive. It lies entirely to the right of the vertical axis.

In the complex plane C\mathbb{C}, the open right half-plane is the set {z∈C:Re⁡(z)>0}\{z \in \mathbb{C} : \operatorname{Re}(z) > 0\}. The closed right half-plane includes the imaginary axis: {z∈C:Re⁡(z)≥0}\{z \in \mathbb{C} : \operatorname{Re}(z) \geq 0\}. Equivalently, in R2\mathbb{R}^2, it is the set of points (x,y)(x, y) with x>0x > 0 (open) or x≥0x \geq 0 (closed).

Key Formula

HR={z∈C:Re⁡(z)>0}\mathbb{H}_R = \{ z \in \mathbb{C} : \operatorname{Re}(z) > 0 \}
Where:
  • zz = A complex number $z = a + bi$
  • Re⁡(z)\operatorname{Re}(z) = The real part of $z$, equal to $a$

How It Works

To determine whether a complex number z=a+biz = a + bi lies in the right half-plane, check whether its real part aa is positive. If a>0a > 0, the point is in the open right half-plane. If a=0a = 0, it sits on the boundary (the imaginary axis). The left half-plane, by contrast, contains all points with Re⁡(z)<0\operatorname{Re}(z) < 0. In control theory and stability analysis, a system is stable when all poles of its transfer function lie in the left half-plane; poles in the right half-plane indicate instability.

Worked Example

Problem: Determine which of the following complex numbers lie in the open right half-plane: z1=3+4iz_1 = 3 + 4i, z2=−2+7iz_2 = -2 + 7i, z3=0+5iz_3 = 0 + 5i.
Step 1: Find the real part of each number.
Re⁡(z1)=3,Re⁡(z2)=−2,Re⁡(z3)=0\operatorname{Re}(z_1) = 3, \quad \operatorname{Re}(z_2) = -2, \quad \operatorname{Re}(z_3) = 0
Step 2: Check the condition Re⁡(z)>0\operatorname{Re}(z) > 0 for each.
3>0  ✓,−2>0  ×,0>0  ×3 > 0 \;\checkmark, \quad -2 > 0 \;\times, \quad 0 > 0 \;\times
Answer: Only z1=3+4iz_1 = 3 + 4i lies in the open right half-plane. The point z3=5iz_3 = 5i lies on the boundary (imaginary axis), and z2z_2 is in the left half-plane.

Why It Matters

In differential equations and control engineering, the location of eigenvalues or poles relative to the right half-plane determines system stability. Laplace transform techniques map time-domain signals into the right half-plane, making it central to signal processing and circuit analysis.

Common Mistakes

Mistake: Confusing the open and closed right half-plane, especially including or excluding the imaginary axis.
Correction: The open right half-plane requires Re⁡(z)>0\operatorname{Re}(z) > 0 (boundary excluded), while the closed version uses Re⁡(z)≥0\operatorname{Re}(z) \geq 0 (boundary included). In stability analysis, the distinction matters because poles exactly on the imaginary axis represent marginal stability, not full stability.