Resurgence of the Airy function and other exponential integrals

To-do list

Conventions

• (Veronica) Change the computation with Sauzin to be in line with convention
• (Aaron) Finish switching the Airy and Airy-Lucas sections to the literature sign convention

Proofreading

• (Aaron) Look over "Why does Borel summation work for solutions of certain level-1 ODEs?"
• (Aaron) Understand new material in "A new perspective: Borel regularity"
• (Aaron) Proofread copied-in sections about Laplace transform
• (Aaron) Check the Bessel equation in the spatial domain that corresponds to the Airy-Lucas equation in the frequency domain
• (Aaron) Look over general modified Bessel example

Introduction

• (Aaron) Draw position domain picture for Watson-Nevanlinna-Sokal theorem.
• (Veronica) Turn Nevanlinna-Watson bullet points into prose

General results

• Prove Theorem 2 (from outline)
• Ask Fredrick whether existence and uniqueness appendix might be publishable on its own
  • (Veronica) Get in touch
  • (Aaron) Send summary of existence and uniqueness results

Examples

• Airy-Lucas
  • (Aaron) Plot Λ^(k) contours
  • (Aaron) Copy the hypergeomtric function identity used in the contour argument into an appendix
  • (Veronica) Copy in asymptotic analysis
  • Show how the proof of Theorem 2 works by going through it in this special case
    • (Veronica) Construct W̃₁ and W̃₋₁ in the "Asymptotic analysis" section
    • (Aaron) Revise the "Borel regularity" section to mirror the proof of Theorem 2
    • (Aaron) Confirm the uniqueness argument by showing explicitly that ṽ₁ sums to a multiple of v₁
• General modified Bessel
  • (Aaron) Plot contours
  • (unassigned) Generalize the differential equation argument used for Airy-Lucas
• (Aaron) Add vibrating beam example
• Fix mistaken coefficients in hyper-Airy 5 example
  • (Aaron) Check over the Sage notebook that gave the coefficients in the current draft
  • (both) Compare with the Mathematica notebook that gives different coefficients
• Generalized Airy
  • (Veronica) Try to do contour argument
• Hyper-Airy
  • Try to do contour argument (stuck right now)
  • Try plugging ζ = 27u⁴ − 4u into a generic hypergeometric series in ζ and comparing with the power series of 1 / 4(27u³ − 1)

Comparisons

• (Veronica) Write out the comparison with Mariño in the desired way

Changes from draft for Tu

• The sentence that starts "The Borel transform maps..."
• Thimble integral proof