Laplace contour integrals and linear differential equations
Abstract
The purpose of this paper is to determine the main properties of Laplace contour integrals $$\Lambda(z)=\frac1{2\pi i}\int_\CC\phi_L(t)e^{-zt}\,dt,$$ that solve linear differential equations $$L[w](z):=w^{(n)}+\sum_{j=0}^{n-1}(a_j+b_jz)w^{(j)}=0.$$ This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén-Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.07550
- arXiv:
- arXiv:2009.07550
- Bibcode:
- 2020arXiv200907550S
- Keywords:
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- Mathematics - Complex Variables