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Concept
Circle group
Summary
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers
\mathbb T = { z \in \mathbb C : |z| = 1 }.
The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well.
A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta:
\theta \mapsto z = e^{i\theta} = \cos\theta + i\sin\theta.
This is the exponential map for the circle group.
The circle group plays a central role in Pontryagin duality and in the theory of Lie groups.
The notation \mathbb T for the ci
Official source
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