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研究生:黃智駿
研究生(外文):HUANG,JHIN-JHUN
論文名稱:廣義(p,q)-三角函數的研究
論文名稱(外文):Topics on Generalized (p,q)-Trigonometric Function
指導教授:鄭彥修
指導教授(外文):CHENG,YAN-HSIOU
口試委員:王惟權劉宣谷
口試委員(外文):WANG,WEI-CHUANLIU,HSUAN-KU
口試日期:2016-06-16
學位類別:碩士
校院名稱:國立臺北教育大學
系所名稱:數學暨資訊教育學系(含數學教育碩士班)
學門:教育學門
學類:普通科目教育學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:中文
論文頁數:28
中文關鍵詞:廣義三角函數廣義圓周率
外文關鍵詞:generalized trigonometric functionsgeneralized pi
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在這篇論文裡,我們首先探討了S_{p}及pi_{p}的函數值隨著p值增加的變化趨勢,其中S_{p}為下列積分之反函數
x=\int^{S_{p}(x)}_{0}(1-|t|^{p})^{-\frac{1}{p}}dt, p>1
且pi_{p}=2\int^{1}_{0}(1-t^{p})^{-\frac{1}{p}}dt。
我們修正了Binding的錯誤,重新證明S_{p}(x)及pi_{p} 對p的單調性。
在本論文的第二部分,我們將S_{p}與pi_{p}的定義推廣至S_{p,q}函數與pi_{p,q},探討其相關微分方程及對p與q的單調性,此處S_{p,q}為下列積分之反函數x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt, p,q>1
且pi_{p,q}=2\int^{1}_{0}(1-t^{q})^{-\frac{1}{p}}dt。
我們發現S_{p,q}與S_{p}、sin(x)有類似的性質,並且證明了下列等價條件
(a)\forall x\in[-\frac{ i_{p,q}}{2},\frac{ i_{p,q}}{2}]

x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt.\forall x \in \mathbb{R}
S_{p,q}(x+k i_{p,q})=(-1)^{k}S_{p,q}(x).

(b)S_{p,q}(0)=0,S'_{p,q}(0)=1
|S_{p,q}|^{q}+|S'_{p,q}|^{p}=1
(c)S_{p,q}(0)=0,S'_{p,q}(0)=1
(|S'_{p,q}|^{p-2}S'_{p,q})'+\frac{q(p-1)}{p}|S_{p,q}|^{q-2}S_{p,q}=0

最後,我們還證明了pi_{p,q}與S_{p,q}( i_{p,q}t),\forall t\in[0,1]的函數值隨著p,q增加而遞減。


In this thesis, we firstly discuss the monotonicity of S_{p}(x) and pi_{p} in p. Here S_{p} is the inverse function of the integral
x=\int^{S_{p}(x)}_{0}(1-|t|^{p})^{-\frac{1}{p}}dt, p>1
and pi_{p}=2\int^{1}_{0}(1-t^{p})^{-\frac{1}{p}}dt.
We revise the mistake that Binding made ,and prove the function S_{p} and pi_{p} decrease in p.
In the second part, we extend the definitions of S_{p} and pi_{p} to S_{p,q} and pi_{p,q}, and discuss the relative differential equations and the monotonicity in p and q. Here S_{p,q} is the inverse function of the integral x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt ,p,q>1
and pi_{p,q}=2\int^{1}_{0}(1-t^{q})^{-\frac{1}{p}}dt. We find that S_{p,q}, S_{p} and sin(x) have similar property, and prove the following equivalent conditions
\begin{enumerate}

(a)\forall x\in[-\frac{ i_{p,q}}{2},\frac{ i_{p,q}}{2}]

x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt.\forall x \in \mathbb{R}
S_{p,q}(x+k i_{p,q})=(-1)^{k}S_{p,q}(x).

(b)S_{p,q}(0)=0,S'_{p,q}(0)=1
|S_{p,q}|^{q}+|S'_{p,q}|^{p}=1
(c)S_{p,q}(0)=0,S'_{p,q}(0)=1
(|S'_{p,q}|^{p-2}S'_{p,q})'+\frac{q(p-1)}{p}|S_{p,q}|^{q-2}S_{p,q}=0

Finally, we also prove that pi_{p,q} and S_{p,q}( i_{p,q}t),\forall t\in[0,1] are decreasing in p,q.

1 Introduction......................... 1
2 pi_{p}和S_{p}的單調性..................7
2.1 pi_{p}單調性.........................8
2.2 S_{p}單調性......................... 9
3 Proof of Theorem 1.2 .................12
4 pi_{p,q}與 S_{p,q} 的性質..............16
4.1 pi_{p,q}的極值...................... 17
4.2 pi_{p,q}對p,q的單調性................20
4.3 S_{p}對p,q的單調性...................21

[1] P. Binding, L. Boulton, J. Čepička, P. Drábek and P. Girg, Basic properties of eigenfunctions of the p Laplacian, Proceedings of The American Mathematical Society (2006), Vol. 134, No. 12, 3487 3494.
[2] C. Bennewitz and Y. Saitō, An embedding norm and the Lindqvist trigonometric functions, Electronic Journal of Differential Equations (2002), No. 86, 1-6.
[3] B. A. Bhayo and M. Vuorinen, On generalized trigonometric functions with two parameters, Journal of Approximation Theory (2012), Vol. 164, Issue 10, 1415-1426
[4] P. Drábek and R. Manásevich, On the closed solution to some p-Laplacian nonhomogeneous eigenvalue problems, Differential Integral Equations 12 (1999), no. 6, 773-788.
[5] 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.
[6] J. Stewart, “Calculus : concepts and contexts”(2003), Belmont, CA : Brooks/Cole, Cengage Learning, pp.477 483.
[7] D. Wei, Y. Liu and M.B. Elgindi, Some generalized trigonometric sine functions and their applications, Applied Mathematical Sciences (2012), Vol. 6, No. 122,
6053-6068.
[8] R. L. Wheeden and A. Zygmund,“Measure and integral, an introduction to real analysis, second edition”, Marcel Dekker, (1977).
[9] 陳惠瑜。廣義三角函數的研究 (2009)。國立中山大學應用數學系研究所碩士論文。
[10] 楊喻文。廣義三角函數及雙曲函數的專題研究 (2013)。國立中山大學應用數學系研究所碩士論文。

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