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論文名稱 Title |
廣義三角函數的研究 On generalized trigonometric functions |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
57 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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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 |
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統計 Statistics |
本論文已被瀏覽 5766 次,被下載 2204 次 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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