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題名 半純函數體中的函數方程
On Functional Equations in the Field of Meromorphic Functions
作者 葉長青
Yeh, Chang Ching
貢獻者 陳天進
Chen, Ten Ging
葉長青
Yeh, Chang Ching
關鍵詞 半純函數
值分佈理論
函數方程
meromorphic function
value distribution theory
functional equation
日期 2009
上傳時間 8-十二月-2010 11:52:40 (UTC+8)
摘要 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.
參考文獻 [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.
[5] F. Gross, On the functional equation fn + gn = hn, Amer. Math. Monthly, 73 (1966), pp. 1093–1096.
[6] F. Gross, Factorization of meromorphic functions, Mathematics Research Center, Naval Research Laboratory, Washington, D. C., 1972.
[7] G. G. Gundersen, Meromorphic solutions of f6 + g6 + h6 ≡ 1, Analysis (Munich), 18 (1998), pp. 285–290.
[8] G. G. Gundersen, Meromorphic solutions of f5 + g5 + h5 ≡ 1, Complex Variables Theory Appl., 43 (2001), pp. 293–298. The Chuang special issue.
[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.
p
[21] N. Toda, On the functional equation
描述 碩士
國立政治大學
應用數學研究所
97751003
98
資料來源 http://thesis.lib.nccu.edu.tw/record/#G0097751003
資料類型 thesis
dc.contributor.advisor 陳天進zh_TW
dc.contributor.advisor Chen, Ten Gingen_US
dc.contributor.author (作者) 葉長青zh_TW
dc.contributor.author (作者) Yeh, Chang Chingen_US
dc.creator (作者) 葉長青zh_TW
dc.creator (作者) Yeh, Chang Chingen_US
dc.date (日期) 2009en_US
dc.date.accessioned 8-十二月-2010 11:52:40 (UTC+8)-
dc.date.available 8-十二月-2010 11:52:40 (UTC+8)-
dc.date.issued (上傳時間) 8-十二月-2010 11:52:40 (UTC+8)-
dc.identifier (其他 識別碼) G0097751003en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/49458-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 應用數學研究所zh_TW
dc.description (描述) 97751003zh_TW
dc.description (描述) 98zh_TW
dc.description.abstract (摘要) 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。		zh_TW
dc.description.abstract (摘要) 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.		en_US
dc.description.tableofcontents 謝辭...............i
Abstract...............ii
中文摘要...............iii
1 Introduction...............1
2 Some Special Functional Equations...............4
3 Basic Theory of Value Distribution...............7
4 Some Necessary Conditions for the Existence of Solution of Functional Equations...............15
5 The Distribution of Zeros and Poles of a Class of Meromorphic Functions...............32
References...............38		zh_TW
dc.format.extent 674396 bytes-
dc.format.mimetype application/pdf-
dc.language.iso en_US-
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0097751003en_US
dc.subject (關鍵詞) 半純函數zh_TW
dc.subject (關鍵詞) 值分佈理論zh_TW
dc.subject (關鍵詞) 函數方程zh_TW
dc.subject (關鍵詞) meromorphic functionen_US
dc.subject (關鍵詞) value distribution theoryen_US
dc.subject (關鍵詞) functional equationen_US
dc.title (題名) 半純函數體中的函數方程zh_TW
dc.title (題名) On Functional Equations in the Field of Meromorphic Functionsen_US
dc.type (資料類型) thesisen
dc.relation.reference (參考文獻) [1] I. N. Baker, On a class of meromorphic functions, Proc. Amer. Math. Soc., 17 (1966), pp. 819–822.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [3] M. L. Green, Some Picard theorems for holomorphic maps to algebraic vari- eties, Amer. J. Math., 97 (1975), pp. 43–75.zh_TW
dc.relation.reference (參考文獻) [4] F. Gross, On the equation fn + gn = 1, Bull. Amer. Math. Soc., 72 (1966), pp. 86–88.zh_TW
dc.relation.reference (參考文獻) [5] F. Gross, On the functional equation fn + gn = hn, Amer. Math. Monthly, 73 (1966), pp. 1093–1096.zh_TW
dc.relation.reference (參考文獻) [6] F. Gross, Factorization of meromorphic functions, Mathematics Research Center, Naval Research Laboratory, Washington, D. C., 1972.zh_TW
dc.relation.reference (參考文獻) [7] G. G. Gundersen, Meromorphic solutions of f6 + g6 + h6 ≡ 1, Analysis (Munich), 18 (1998), pp. 285–290.zh_TW
dc.relation.reference (參考文獻) [8] G. G. Gundersen, Meromorphic solutions of f5 + g5 + h5 ≡ 1, Complex Variables Theory Appl., 43 (2001), pp. 293–298. The Chuang special issue.zh_TW
dc.relation.reference (參考文獻) [9] W. Hayman, Warings Problem fu ̈r analytische Funktionen, Bayer. Akad. Wiss. Math.-Natur. Kl. Sitzungsber., (1984), pp. 1–13 (1985).zh_TW
dc.relation.reference (參考文獻) [10] W. K. Hayman, Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [12] A. V. Jategaonkar, Elementary proof of a theorem of P. Montel on entire functions, J. London Math. Soc., 40 (1965), pp. 166–170.zh_TW
dc.relation.reference (參考文獻) [13] I. Lahiri and K.-W. Yu, On generalized Fermat type functional equations, Comput. Methods Funct. Theory, 7 (2007), pp. 141–149.zh_TW
dc.relation.reference (參考文獻) [14] D. H. Lehmer, On the diophantine equation x3 + y3 + z3 = 1, Journal of the London Mathematical Society, 31 (1956), pp. 275–280.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [17] P. Montel, Le ̧cons sur les familles normales de fonctions analytiques et leurs applications, Gauthiers-Villars, Paris, 1927.zh_TW
dc.relation.reference (參考文獻) [18] R. Nevanlinna, Le th ́eor"eme de Picard-Borel et la th ́eorie des fonctions m ́eromorphes, Gauthiers-Villars, Paris, 1929.zh_TW
dc.relation.reference (參考文獻) [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.zh_TW
dc.relation.reference (參考文獻) [20] F. Rellich, Elliptische Funktionen und die ganzen L ̈osungen von y′′ = f(y) , Math, 47 (1940), pp. 153–160.zh_TW
dc.relation.reference (參考文獻) pzh_TW
dc.relation.reference (參考文獻) [21] N. Toda, On the functional equationzh_TW