2000 Fiscal Year Final Research Report Summary
a study of special functions defined by functional equations
| Project/Area Number |
11640212
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Keio University |
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Principal Investigator |
SHIMOMURA Shun Keio University Mathematics, Associate Professor, 理工学部, 助教授 (00154328)
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| Co-Investigator(Kenkyū-buntansha) |
TANI Atusi Keio University Professor, 理工学部, 教授 (90118969)
SHIOKAWA Iekata Keio University Professor, 理工学部, 教授 (00015835)
KIKUCHIO Norio Keio University Professor, 理工学部, 教授 (80090041)
NAKANO Minoru Keio University Assistant Professor, 理工学部, 講師 (00051607)
ISHIKAWA Shiro Keio University Associate Professor, 理工学部, 助教授 (10051913)
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| Project Period (FY) |
1999 – 2000
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| Keywords | confluent hypergeometric function / asymptotic expansion / Painleve equation / value distribution / growth order / Garnier系 |
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| Research Abstract |
1. We clarified the asymptotic behavior of a confluent hypergeometric function by using an integral representation. Applying the result we also examined the global behavior of solutions of a confluent Pochhammer equation.
2. We examined the value distribution of Painleve transcendents of the third and the fifth kind. For Painleve transcendents of the first, the second and the fourth kind, we proved the finiteness of the growth order by two different methods. For these Painleve transcendents we examined the deficiency of small functions.
3. We proved the Painleve property of a degenerate Garnier system which is a two-variable version of the first Painleve equation, and also proved that, for every solution of it, the singular loci are analytic sets expressed in terms of solutions of a fourth-order nonlinear ordinary differential equation.
4. For linear differential equations with doubly periodic meromorphic coefficients, we examined the value distribution and the growth order of meromorphic solutions. For Riccati differential equations with elliptic coefficients, we proved that, under certain conditions, every periodic solution is doubly periodic, and obtained expressions of such solutions.
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