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URN etd-0625110-112752
Author Hui-yu Chen
Author's Email Address No Public.
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Department Applied Mathematics
Year 2009
Semester 2
Degree Master
Type of Document
Language English
Title On generalized trigonometric functions
Date of Defense 2009-10-30
Page Count 57
Keyword
  • generalized trigonometric functions
  • generalized sine function
  • identities
  • p-Laplacian
  • 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.
    Advisory Committee
  • Wei-Cheng Lian - chair
  • Yung-sze Choi - co-chair
  • Yan-hsiou Cheng - co-chair
  • Chun-kong Law - advisor
  • Files
  • etd-0625110-112752.pdf
  • indicate accessible in a year
    Date of Submission 2010-06-25

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