English
For a complex algebra A with a ℂ-algebra structure, SpectrumRestricts a Complex.reCLM holds iff every spectral value x in spectrum ℂ a is real (x = Complex.ofReal x.re).
Русский
Для алгебры A над комплексами с структурой ℂ-алгебры, SpectrumRestricts a Complex.reCLM выполняется тогда и только тогда, когда каждое значение спектра x ∈ spectrum ℂ a реально (x = Complex.ofReal x.re).
LaTeX
$$$\text{SpectrumRestricts } a \ Complex.reCLM \iff \forall x \in \mathrm{spectrum}\_\mathbb{C}(a), x = x_{\mathrm{re}}$$$
Lean4
/-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
theorem taylorSeries_eq_of_entire : ∑' n : ℕ, (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c = f z :=
(hasSum_taylorSeries_of_entire hf c z).tsum_eq
