ECE 486 REVIEW: COMPLEX VARIABLES Fall 08
Reading: FPE, Appendix B.
Imaginary operator: j, defined such that j2=−1.
Complex numbers. A complex number sis of the form s=σ+jω, where σand ωare real numbers. The
number σis called the real part of s, and is denoted by σ= Re(s). The number ωis called the imaginary
part of sand is denoted by ω= Im(s).
Complex plane: A geometric representation of complex numbers, with a real axis (horizontal) and an
imaginary axis (vertical). The real axis represents all complex numbers ssuch that Im(s) = 0. The
imaginary axis represents all complex numbers ssuch that Re(s) = 0. The following regions of the plane
will be useful to our discussions.
•All complex numbers ssatisfying Re(s)<0 are said to lie in the Open Left Half Plane (OLHP).
•All complex numbers ssatisfying Re(s)≤0 are said to lie in the Closed Left Half Plane (CLHP).
•All complex numbers ssatisfying Re(s)>0 are said to lie in the Open Right Half Plane (ORHP).
•All complex numbers ssatisfying Re(s)≥0 are said to lie in the Closed Right Half Plane (CRHP).
Figure 1: A complex number s=σ+jω in the complex plane.
Polar Form. Complex numbers can also be represented in the polar form s=rejθ , where
r=pσ2+ω2, θ = arctan ω
σ.
The number ris called the magnitude of s, and is sometimes denoted by r=|s|. The number θis called
the phase of s(in radians), and is denoted by θ=∠s. The number rrepresents the length of the vector
corresponding to the point σ+jω, and θrepresents its angle from the positive real axis.
