OAI/MathematicalPhysics/DefocusingNLS/Profile/SlowKernelDerivative.lean

GitHub source file

6 declarations, 0 contain sorry, 0 declared as axioms

1. theorem OAI.DefocusingNLS.one_add_real_div_mem_slitPlane line 9
2. def OAI.DefocusingNLS.slowKernelSpatialDerivative line 19
3. theorem OAI.DefocusingNLS.hasDerivAt_regularizingBracket_spatial line 25
4. theorem OAI.DefocusingNLS.hasDerivAt_regularizedSlowKernel_spatial line 36
5. theorem OAI.DefocusingNLS.slowKernelSpatialDerivative_eq line 48

   Source documentation

   The spatial derivative is a shifted regularized kernel plus the gamma integrand. This retains the endpoint cancellation instead of separating two individually divergent terms at the original parameter.

   theorem slowKernelSpatialDerivative_eq (q : ℂ) (m : ℕ) (x : ℂ)
       {u : ℝ} (hu : 0 < u) :
       slowKernelSpatialDerivative q m x u =
         (-((m : ℂ) - 1 - q) / x ^ 2) *
           (regularizedSlowKernel (q + 1) m x u +
             Complex.exp (-(u : ℂ)) * (u : ℂ) ^ q) := by
   …

   Mentions: OAI.DefocusingNLS.slowKernelSpatialDerivative, OAI.DefocusingNLS.regularizedSlowKernel

6. theorem OAI.DefocusingNLS.integrable_slowKernelSpatialDerivative line 70