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| Title: | 半純函數體中的函數方程
On Functional Equations in the Field of Meromorphic Functions |
| Authors: | 葉長青
Yeh, Chang Ching |
| Contributors: | 陳天進
Chen, Ten Ging
葉長青
Yeh, Chang Ching |
| Keywords: | 半純函數
值分佈理論
函數方程
meromorphic function
value distribution theory
functional equation |
| Date: | 2009 |
| Issue Date: | 2010-12-08 11:52:40 (UTC+8) |
| Abstract: | 在這篇論文中，我們將利用值分佈的理論來探討下列函數方程解的存在性與其性質：
\\[\\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\\]
其中 $a_1(z),\\cdots ,a_p(z)$ 為半純函數。對某些特殊方程，除了文獻裡已知的結果外，我們亦提供其它的例子。一般而言，我們探討解存在的必要條件。另外，我們證明了某一類半純函數之零點與極點之分佈的結果。
In this thesis, we use the theory of value distribution to study the existence of solution of the following functional equation:
\\[\\sum_{j=1}^pa_j(z)f_j(z)^{k_j}=1,\\]
where $a_1(z),\\cdots ,a_p(z)$ are meromorphic functions. For some special case, new and old examples of the solutions are given. For the general case, a necessary condition for the existence of solution is considered. Moreover, we obtain a result on the distribution of zeros and poles of a class of meromorphic functions. |
| Reference: | [1] I. N. Baker, On a class of meromorphic functions, Proc. Amer. Math. Soc., 17 (1966), pp. 819–822.
[2] C.-T. Chuang and C.-C. Yang, Fix-points and factorization of meromor- phic functions, World Scientific Publishing Co. Inc., Teaneck, NJ, 1990. Trans- lated from the Chinese.
[3] M. L. Green, Some Picard theorems for holomorphic maps to algebraic vari- eties, Amer. J. Math., 97 (1975), pp. 43–75.
[4] F. Gross, On the equation fn + gn = 1, Bull. Amer. Math. Soc., 72 (1966), pp. 86–88.
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[9] W. Hayman, Warings Problem fu ̈r analytische Funktionen, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., (1984), pp. 1–13 (1985).
[10] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.
[11] K. Ishizaki, A note on the functional equation fn+gn+hn = 1 and some com- plex differential equations, Comput. Methods Funct. Theory, 2 (2002), pp. 67– 85.
[12] A. V. Jategaonkar, Elementary proof of a theorem of P. Montel on entire functions, J. London Math. Soc., 40 (1965), pp. 166–170.
[13] I. Lahiri and K.-W. Yu, On generalized Fermat type functional equations, Comput. Methods Funct. Theory, 7 (2007), pp. 141–149.
[14] D. H. Lehmer, On the diophantine equation x3 + y3 + z3 = 1, Journal of the London Mathematical Society, 31 (1956), pp. 275–280.
[15] P. Li and C.-C. Yang, Some further results on the unique range sets of meromorphic functions, Kodai Math. J., 18 (1995), pp. 437–450.
[16] H. Milloux, Les fonctions m ́eromorphes et leurs d ́eriv ́ees. Extensions d’un th ́eor"eme de M. R. Nevanlinna. Applications, Actualit ́es Sci. Ind., no. 888, Hermann et Cie., Paris, 1940.
[17] P. Montel, Le ̧cons sur les familles normales de fonctions analytiques et leurs applications, Gauthiers-Villars, Paris, 1927.
[18] R. Nevanlinna, Le th ́eor"eme de Picard-Borel et la th ́eorie des fonctions m ́eromorphes, Gauthiers-Villars, Paris, 1929.
[19] D. J. Newman and M. Slater, Waring’s problem for the ring of polynomi- als, Journal of Number Theory, 11 (1979), pp. 477–487.
[20] F. Rellich, Elliptische Funktionen und die ganzen L ̈osungen von y′′ = f(y) , Math, 47 (1940), pp. 153–160.
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[21] N. Toda, On the functional equation |
| Description: | 碩士
國立政治大學
應用數學研究所
97751003
98 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0097751003 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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