mathlib documentation

analysis.complex.circle

The circle #

This file defines circle to be the metric sphere (metric.sphere) in centred at 0 of radius 1. We equip it with the following structure:

We furthermore define exp_map_circle to be the natural map λ t, exp (t * I) from to circle, and show that this map is a group homomorphism.

Implementation notes #

Because later (in geometry.manifold.instances.sphere) one wants to equip the circle with a smooth manifold structure borrowed from metric.sphere, the underlying set is {z : ℂ | abs (z - 0) = 1}. This prevents certain algebraic facts from working definitionally -- for example, the circle is not defeq to {z : ℂ | abs z = 1}, which is the kernel of complex.abs considered as a homomorphism from to , nor is it defeq to {z : ℂ | norm_sq z = 1}, which is the kernel of the homomorphism complex.norm_sq from to .

@[simp]
theorem circle_def  :
@[simp]
@[protected, instance]
noncomputable def circle.comm_group  :
Equations
@[simp]
theorem coe_inv_circle (z : circle) :
@[simp]
theorem coe_div_circle (z w : circle) :
(z / w) = z / w
noncomputable def circle.to_units  :

The elements of the circle embed into the units.

Equations
@[protected, instance]
@[simp]
noncomputable def circle.of_conj_div_self (z : ) (hz : z 0) :

If z is a nonzero complex number, then conj z / z belongs to the unit circle.

Equations
noncomputable def exp_map_circle  :

The map λ t, exp (t * I) from to the unit circle in .

Equations
@[simp]
noncomputable def exp_map_circle_hom  :

The map λ t, exp (t * I) from to the unit circle in , considered as a homomorphism of groups.

Equations