| 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. |
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