probChar — Mathlib

English

The additive character 𝐞 is given by x ↦ exp(2π x i) and respects addition: 𝐞(x+y) = 𝐞(x) 𝐞(y).

Русский

Добавочный характер 𝐞 задаётся x ↦ e^{2π i x} и обладает свойством: 𝐞(x+y) = 𝐞(x) 𝐞(y).

LaTeX

$$$$ 𝐞(x) = \\exp\\bigl(2\\pi i x\\bigr), \\quad 𝐞(x+y) = 𝐞(x) \\cdot 𝐞(y). $$$$

Lean4

/-- The additive character from `ℝ` onto the circle, given by `fun x ↦ exp (x * I)`. This uses the
probabilist convention that there is no `2 * π` in the exponent. -/
def probChar : AddChar ℝ Circle where
  toFun := Circle.exp
  map_zero_eq_one' := Circle.exp_zero
  map_add_eq_mul' := Circle.exp_add

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