### Title page for etd-0625110-112752

URN etd-0625110-112752 Hui-yu Chen No Public. This thesis had been viewed 5212 times. Download 1927 times. Applied Mathematics 2009 2 Master English On generalized trigonometric functions 2009-10-30 57 generalized trigonometric functions generalized sine function identities p-Laplacian The function \$sin x\$ as one of the six trigonometric functions isfundamental in nearly every branch of mathematics, and itsapplications. In this thesis, we study an integral equation relatedto 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 propertiesare also similar to those of \$sin x\$ : differentiation, identities,periodicity, asymptotic expansions, \$cdots\$, etc. For example, wehave\$\$|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, wedefine 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}]\$ andderive its properties. Thus we obtain two sets of trigonometricfunctions: 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 functionshave similar differentiation formulas, identities and periodicproperties as the classical trigonometric functions. They coincidewhen \$p=2\$.Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions. Wei-Cheng Lian - chair Yung-sze Choi - co-chair Yan-hsiou Cheng - co-chair Chun-kong Law - advisor indicate accessible in a year 2010-06-25

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