OAI/MathematicalPhysics/DefocusingNLS/Profile/SlowEulerParameter.lean

GitHub source file

9 declarations, 0 contain sorry, 0 declared as axioms

Holomorphic dependence of the Euler integral on its parameter

The logarithm introduced by parameter differentiation is absorbed by a positive shift in the real exponent. The majorant is another convergent Euler integral, so no endpoint logarithmic estimates are needed.

  1. def OAI.DefocusingNLS.slowEulerParameterDerivative line 16
  2. theorem OAI.DefocusingNLS.hasDerivAt_slowEulerKernel_parameter line 19
  3. theorem OAI.DefocusingNLS.norm_slowEulerKernel_parameter_ratio line 39
  4. theorem OAI.DefocusingNLS.norm_slowEulerParameterDerivative line 57
  5. theorem OAI.DefocusingNLS.slowEulerParameterDerivative_majorant line 68
  6. theorem OAI.DefocusingNLS.continuousOn_slowEulerParameterDerivative line 90
  7. theorem OAI.DefocusingNLS.hasDerivAt_slowEulerIntegral_parameter line 103
  8. theorem OAI.DefocusingNLS.differentiableAt_regularizedSlowSolution_parameter line 134

    Source documentation

    Holomorphy in the parameter includes the regularized point q=0.

    theorem differentiableAt_regularizedSlowSolution_parameter (q : ℂ) (m : ℕ) (x : ℂ)
    (hq : -1 < q.re) (hx : 0 < x.re) :
    DifferentiableAt ℂ (fun z : ℂ => regularizedSlowSolution z m x) q := by
…

    Mentions: OAI.DefocusingNLS.regularizedSlowSolution

  9. theorem OAI.DefocusingNLS.differentiableOn_regularizedSlowSolution_parameter line 157