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def definition def or abbrev

kernel_response_trunc

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formal statement (Lean)

 145def kernel_response_trunc (H : CaldeiraLeggett.TransferFunction) (ω B : ℝ) : ℂ :=

proof body

Definition body.

 146  ∫ t in (0 : ℝ)..B,
 147    ((debye_kernel H t : ℝ) : ℂ) * Complex.exp (-(Complex.I * (ω : ℂ) * (t : ℂ)))
 148
 149
 150/-!
 151### Bridge lemma (frequency-domain closed form)
 152
 153For τ>0, define `a = (1/τ) + i ω`. Then
 154
 155  exp(-t/τ) * exp(-i ω t) = exp(-(a * t)).
 156
 157The truncated integral can be evaluated in closed form using `integral_exp_smul_neg`,
 158then the `B → ∞` limit is obtained from `tendsto_exp_neg_mul_ofReal_atTop`.
 159-/
 160
 161/-! ### Laplace transform limit and transfer-function bridge -/
 162
 163/-- The complex exponent `a = (1/τ) + i ω` appearing in the Debye kernel transform. -/

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