Journal of the Mathematical Society of Japan

A functional equation with Borel summable solutions and irregular singular solutions

Sunao ŌUCHI

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Abstract

A Functional equation $\sum_{i=1}^{m}a_{i}(z)u(\varphi_{i}(z))=f(z)$ is considered. First we show the existence of solutions of formal power series. Second we study the homogeneous equation $(f(z)\equiv 0)$ and construct formal solutions containing exponential factors. Finally it is shown that there exists a genuine solution in a sector whose asymptotic expansion is a formal solution, by using the theory of Borel summability of formal power series. The equation has similar properties to those of irregular singular type in the theory of ordinary differential equations.

Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 711-731.

Dates
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1524038671

Digital Object Identifier
doi:10.2969/jmsj/07027491

Mathematical Reviews number (MathSciNet)
MR3787737

Zentralblatt MATH identifier
1395.30031

Subjects
Primary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX]
Secondary: 39B32: Equations for complex functions [See also 30D05] 44A10: Laplace transform

Keywords
Borel summable irregular singular functional equation

Citation

ŌUCHI, Sunao. A functional equation with Borel summable solutions and irregular singular solutions. J. Math. Soc. Japan 70 (2018), no. 2, 711--731. doi:10.2969/jmsj/07027491. https://projecteuclid.org/euclid.jmsj/1524038671


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References

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