Theorems · Theorem · number theory
Complex.tsum_exp_neg_quadratic
∀ {a : ℂ},
0 < a.re →
∀ (b : ℂ),
∑' (n : ℤ), Complex.exp (-↑Real.pi * a * ↑n ^ 2 + 2 * ↑Real.pi * b * ↑n) =
1 / a ^ (1 / 2) * ∑' (n : ℤ), Complex.exp (-↑Real.pi / a * (↑n + Complex.I * b) ^ 2)Jacobi's theta-function transformation formula for the sum of exp -Q(x), where Q is a
negative definite quadratic form.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 294 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
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Cites56
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- mul_oneproof · cited by 3,885
- add_zeroproof · cited by 2,707
- Nat.cast_oneproof · cited by 2,501
- zero_addproof · cited by 2,366
- mul_commproof · cited by 2,262
- SummationFilter.unconditionalstatement and proof · cited by 2,068
- le_reflproof · cited by 2,061
- absproof · cited by 1,814
- Real.pistatement and proof · cited by 1,774
Cited by3
Results whose statement or proof uses this declaration.
- jacobiTheta₂_functional_equationproof · cited by 3
- Real.tsum_exp_neg_mul_int_sqproof · cited by 0
- Complex.tsum_exp_neg_mul_int_sqproof · cited by 0