Complex numbers
In complex analysis, the complex numbers are customarily represented by the symbol z, which can be separated into its real (x) and imaginary (y) parts:
In the Cartesian plane the

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Argand diagram refers to a geometric plot of complex numbers as points z = x + iy using the x-axis as the real axis and y-axis as the imaginary axis. Such plots are named after Jean-Robert Argand (1768–1822), although they were first described by

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It can be useful to think of the complex plane as if it occupied the surface of a sphere. Given a sphere of unit radius, place its center at the origin of the complex plane, oriented so that the equator on the sphere coincides with the unit circle in the plane, and the north pole

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When discussing functions of a complex variable it is often convenient to think of a cut in the complex plane. This idea arises naturally in several

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In control theory, one use of the complex plane is known as the 's-plane'. It is used to visualise the roots of the equation describing a system's

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The complex plane is associated with two distinct quadratic spaces. For a point z = x + iy in the complex plane, the squaring function z

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We have already seen how the relationship
Imagine two copies of the cut complex plane, the cuts extending along the positive real axis from z = 0 to the point at infinity. On one sheet define 0 ≤ arg(z) < 2π, so that 1 = e = 1, by definition. On the second sheet define 2π ≤

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