Theorems · Theorem · special functions
Polynomial.Chebyshev.U_complex_cos
∀ (θ : ℂ) (n : ℤ), Polynomial.eval (Complex.cos θ) (Polynomial.Chebyshev.U ℂ n) * Complex.sin θ = Complex.sin ((↑n + 1) * θ)
The n-th Chebyshev polynomial of the second kind evaluates on cos θ to the
value sin ((n + 1) * θ) / sin θ.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Complexstatement and proof · cited by 5,565
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- mul_assocproof · cited by 1,667
- Polynomial.Xproof · cited by 1,639
- pow_oneproof · cited by 894
- Polynomial.evalstatement and proof · cited by 796
- Int.cast_natCastproof · cited by 393
- Int.cast_oneproof · cited by 371
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Chebyshev.U_complex_coshproof · cited by 2
- Polynomial.Chebyshev.U_real_cosproof · cited by 1
- Polynomial.Chebyshev.S_two_mul_complex_cosproof · cited by 1