The R Euler–Poisson system on the full slit domain

theorem DirichletTransform.carlsonEulerPoissonOperator_regCarlsonRSlit {ι : Type u_1} [Fintype ι] (t : ℂ) (b : ι → ℂ) {z : ι → ℂ} (hz : z ∈ carlsonRSlitDomain) (i j : ι) : carlsonEulerPoissonOperator i j b z (regCarlsonRSlit t b) = 0

The Euler–Poisson system for R, for all complex parameters and slit-plane nodes. Neither distinct indices nor separated nodes are required.

theorem DirichletTransform.carlsonEulerPoissonOperator_carlsonRSlit {ι : Type u_1} [Fintype ι] (t : ℂ) (b : ι → ℂ) {z : ι → ℂ} (hz : z ∈ carlsonRSlitDomain) (i j : ι) : carlsonEulerPoissonOperator i j b z (carlsonRSlit t b) = 0

The ordinary normalization obeys the same Euler–Poisson system. At Gamma poles this is an identity of totalized expressions, not a finite ordinary value.