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博碩士論文 etd-0625110-112752 詳細資訊
Title page for etd-0625110-112752
論文名稱
Title
廣義三角函數的研究
On generalized trigonometric functions
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
57
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2009-10-30
繳交日期
Date of Submission
2010-06-25
關鍵字
Keywords
恆等式、廣義正弦函數、廣義三角函數
generalized trigonometric functions, generalized sine function, identities, p-Laplacian
統計
Statistics
本論文已被瀏覽 5766 次,被下載 2204
The thesis/dissertation has been browsed 5766 times, has been downloaded 2204 times.
中文摘要
作為六個三角函數之一的函數, $sin x$
幾乎在每個數學分支及相關應用都是基礎且重要的。在這篇論文裡,我們研究了與
$sin x$ 相關的積分方程:對於
$p>1mbox{~及~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}]mbox{,}$
$$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt$$
,其中~$hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dtmbox{。}$我們發現
$S_{p}(x)$ 定義是明確的。它與 $sin x$
也有相似的性質,如微分公式、恆等式、週期性與漸進展開式 $cdots$
等等。例如我們有
$$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~和~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x)mbox{~。~}$$
我們稱 $S_{p}(x)$ 為廣義正弦函數。同樣地當
$xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}]$
時,我們從積分方程 $|x|=int_{C_{p}(x)}^{1}(1-
t^{p})^{-frac{1}{p}}dt$ 中定義 $C_{p}(x)$ 為廣義餘弦函數,
並且推導它的性質。如此,我們得到兩套三角函數,它們分別是
egin{itemize}
item[(i)]$~S_{p}(x),~ S'_{p}(x),~
T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~
SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$
item[(ii)]$~C_{p}(x),~
C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~
CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~
CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~。~}$
end{itemize}
這兩套函數就如同典型的三角函數有相似的微分公式、恆等式及週期性。而且,它們在 $p=2$ 時是相同的。此外,它們的圖和漸進展開式也非常有意思。
因此希望藉由這項研究我們更瞭解三角函數的理論架構。
Abstract
The function $sin x$ as one of the six trigonometric functions is
fundamental in nearly every branch of mathematics, and its
applications. In this thesis, we study an integral equation related
to that of $sin x$:
$mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}]
mbox{~and~} p>1$
$$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$
We find that the function $S_{p}(x)$ is well defined. Its properties
are also similar to those of $sin x$ : differentiation, identities,
periodicity, asymptotic expansions, $cdots$, etc. For example, we
have
$$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$
We call $S_{p}(x)$ the generalized sine function. Similarly, we
define the generalized cosine function $C_{p}(x)$ by
$|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for
$xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and
derive its properties. Thus we obtain two sets of trigonometric
functions: egin{itemize}
item[(i)]$~S_{p}(x),~ S'_{p}(x),~
T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~
SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$
item[(ii)]$~C_{p}(x),~
C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~
CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~
CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~。~}$
end{itemize}These two sets of functions
have similar differentiation formulas, identities and periodic
properties as the classical trigonometric functions. They coincide
when $p=2$.
Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.
目次 Table of Contents
1 Introduction 1
2 Generalized sine and cosine functions 5
3 Other trigonometric functions 25
4 Conclusion 37
Bibliography 39
Appendix 41
參考文獻 References
[1]賈乃輝、魏文恩、郭倍綸、劉玠玟、羅春光 (2006) 以微積分方法探討三角函數的性質,數學傳播季刊,第30卷,第3期,42-52.
[2] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, (1972) Dover, New York.
[3] P. Binding, P. Drabek, Sturm-Liouville theory for the p-Laplacian, Studia Scientiarum Math-ematicarum Hungarica, 40(2003), 373-396.
[4] P. Binding, L. Boulton, J. Cepicka, P. Drabek, and P. Girg, Basis properties of eigenfunctions of the p-Laplacian, Proceedings of the American Mathematical Society, (2006), 3487-3494.
[5] G. Birkhff and G.C. Rota, Ordinary Differential Equations, 4th ed (1989) Wiley, New York.
[6] A. Elbert, A half-linear second order differential equation, Colloqia Mathematica Societatis Jonos Bolyai, 30 Qualitative Theory of Differential Equations, Szeged (Hungary) (1979), 153-180.
[7] C.K. Law, W.C. Lian and W.C. Wang, The inverse nodal problem and the Ambarzumyan problem for the p-Laplacian, Proceedings of the Royal Society of Edinburgh, 139A (2009),
1261-1273.
[8] N.N. Lebedev, R.A. Silverman, Special Functions and Their Applications, (1972) Dover, New York.
[9] P. Lindqvist, Some remarkable sine and cosine functions, Ricerche di Matematica, XLIV, fasc 2 ( 1995), 269-290.
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