Created the Real Analysis Theorems and Definitions packet
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\chapter{Sequences of Functions}
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\subimport{./}{pointwise-and-uniform-convergence.tex}
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\subimport{./}{interchange-of-limits.tex}
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\subimport{./}{the-exponential-and-logarithmic-functions.tex}
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\subimport{./}{the-trigonometric-functions.tex}
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\section{Interchange of Limits}
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\begin{theorem}
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Let $(f_n)$ be a sequence of continuous functions on a set $A \subseteq \R$ and suppose that $(f_n)$ converges uniformly on $A$ to a function $f:A \to \R$. Then $f$ is continuous on $A$.
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\end{theorem}
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\begin{remark}
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Although the uniform convergence of the sequence of continuous functions is sufficient to guarantee the continuity of the limit function, it is \textit{not} necessary.
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\end{remark}
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\begin{theorem}
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Let $J \subseteq \R$ be a bounded interval and let $(f_n)$ be a sequence of functions on $J$ to $\R$. Suppose that there exists $x_0 \in J$ such that $(f_n(x_0))$ converges, and that the sequence $(f'_n)$ of derivatives exists on $J$ and converges uniformly on $J$ to a function $g$.
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\\Then the sequence $(f_n)$ converges uniformly on $J$ to a function $f$ that has a derivative at every point of $J$ and $f'=g$.
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\end{theorem}
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\begin{theorem}
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Let $(f_n)$ be a sequence of functions in $\mathcal{R}[a,b]$ and suppose that $(f_n)$ converges \textbf{uniformly} on $[a,b]$ to $f$. Then $f \in \mathcal{R}[a,b]$ and
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\[\displaystyle\int_{a}^{b}f=\lim\limits_{n \to \infty}\displaystyle\int_{a}^{b}f_n\]
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holds.
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\end{theorem}
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\begin{theorem}[\textbf{Bounded Convergence Theorem}]
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Let $(f_n)$ be a sequence in $\mathcal{R}[a,b]$ that converges on $[a,b]$ to a function $f \in \mathcal{R}[a,b]$.. Suppose also that there exists $B >0$ such that $|f_n(x)|\leq B$ for all $x \in [a,b]$, $n \in \N$. Then
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\[\displaystyle\int_{a}^{b}f=\lim\limits_{n \to \infty}\displaystyle\int_{a}^{b} f_n\]
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holds.
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\end{theorem}
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\begin{theorem}[\textbf{Dini's Theorem}]
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Suppose that $(f_n)$ is a monotone sequence of continuous functions on $I:=[a,b]$ that converges on $I$ to a continuous function $f$. Then the convergence of the sequence is uniform.
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\end{theorem}
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\section{Pointwise and Uniform Convergence}
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\begin{definition}
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Let $(f_n)$ be a sequence of functions on $A \subseteq \R$ to $\R$, let $A_0\subseteq A$, and let $f: A_0 \to \R$. We say that the \textbf{sequence $(f_n)$ converges on $A_0$ to $f$} if, for each, $x \in A_0$, the sequence $(f_n(x))$ converges to $f(x)$ in $\R$. In this case we call $f$ the \textbf{limit on $A_0$ of the sequence $(f_n)$}. When such a function $f$ exists, we say that the sequence $(f_n)$ \textbf{is convergent on $A_0$}, or that $(f_n)$ \textbf{converges pointwise on $A_0$}.
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\end{definition}
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\begin{lemma}
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A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ converges to a function $f:A_0 \to \R$ on $A_0$ if and only if for each $\varepsilon>0$ and each $x \in A_0$ there is a natural number $K(\varepsilon, x)$ such that if $n \geq K(\varepsilon, x)$, then
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\[|f_n(x)-f(x)|<\varepsilon\]
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\end{lemma}
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\begin{definition}
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A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ \textbf{converges uniformly on $A_0 \subseteq A$} to a function $f:A_0 \to \R$ if for each $\varepsilon >0$ there is a natural number $K(\varepsilon)$ (depending on $\varepsilon$ but \textbf{not} on $x \in A_0$) such that if $n \geq K(\varepsilon)$, then
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\[|f_n(x)-f(x)|<\varepsilon\ \forall\ x \in A_0\]
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In this case we say that the sequence $(f_n)$ is \textbf{uniformly convergent on $A_0$}. Sometimes we write
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\[f_n \rightrightarrows f\ \text{on}\ A_0\ \text{or}\ f_n(x)\rightrightarrows f(x)\ \text{for}\ x \in A_0\]
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\end{definition}
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\begin{lemma}
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A sequence $(f_n)$ of functions on $A \subseteq \R$ to $\R$ does not converge uniformly on $A_0 \subseteq A$ to a function $f:A_0 \to \R$ if and only if for some $\varepsilon_0 >0$ there is a subsequence $(f_{n_k})$ of $(f_n)$ and a sequence $(x_k)$ in $A_0$ such that
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\[|f_{n_k}(x_k)-f(x_k)|\geq\varepsilon_0\ \forall\ k \in \N\]
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\end{lemma}
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\begin{definition}
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If $A \subseteq \R$ and $\varphi : A \to \R$ is a function, we say that $\varphi$ is \textbf{bounded on $A$} if the set $\varphi(A)$ is a bounded subset of $\R$. If $\varphi$ is bounded we define the \textbf{uniform norm of $\varphi$ on $A$} by
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\[||\varphi||_A:=\sup\{|\varphi(x)|:x \in A\}\]
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Note that it follows that if $\varepsilon >0$, then
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\[||\varphi||_A \leq \varepsilon \iff |\varphi(x)|\leq \varepsilon\ \forall\ x \in A\]
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\end{definition}
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\begin{lemma}
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A sequence $(f_n)$ of bounded functions on $A\subseteq \R$ converges uniformly on $A$ to $f$ if and only if $||f_n - f||_A \to 0$.
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\end{lemma}
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\begin{theorem}[\textbf{Cauchy Criterion for Uniform Convergence}]
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Let $(f_n)$ be a sequence of bounded functions on $A \subseteq \R$. Then this sequence converges uniformly on $A$ to a bounded function $f$ if and only if for each $\varepsilon>0$ there is a number $H(\varepsilon) \in \N$ such that for all $m,n\geq H(\varepsilon)$, then $||f_m-f_n||_A \leq \varepsilon$.
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\end{theorem}
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\section{The Exponential and Logarithmic Functions}
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\begin{theorem}
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There exists a function $:\R \to \R$ such that:
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\begin{enumerate}
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\item $E'(x)=E(x)\ \forall\ x \in \R$.
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\item $E(0)=1$.
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\end{enumerate}
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\end{theorem}
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\begin{corollary}
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The function $E$ has a derivative of every order and $E^{(n)}(x)=E(x)$ for all $n \in \N$, $x \in \R$.
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\end{corollary}
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\begin{corollary}
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If $x>0$, then $1+x < E(x)$.
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\end{corollary}
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\begin{theorem}
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The function $E:\R\to\R$ that satisfies (1) and (2) of \textit{Theorem 8.3.1} is unique.
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\end{theorem}
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\begin{theorem}
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The unique function $E:\R\to\R$, such that $E'(x)=E(x)$ for all $x \in \R$ and $E(0)=1$, is called the \textbf{exponential function}. The number $e=E(1)$ is called \textbf{Euler's number}. We will frequently write
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\[\exp(x):=E(x)\ \text{or}\ e^x:=E(x)\ \text{for}\ x \in \R\]
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\end{theorem}
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\begin{theorem}
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The exponential function satisfies the following properties:
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\begin{enumerate}
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\item $E(x) \neq 0$ for all $x \in \R$;
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\item $E(x_+y)=E(x)E(y)$ for all $x,y,\in\R$.
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\item $E(r) = e^r$ for all $r \in \Q$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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The exponential function $E$ is strictly increasing on $\R$ and has range equal to $\{y \in \R : y > 0\}$. Further, we have
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\[\lim\limits_{x \to -\infty} E(x)=0\ \ \text{and}\ \ \lim\limits_{x \to \infty} = \infty\]
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\end{theorem}
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\begin{definition}
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The function inverse to $E:\R \to \R$ is called the \textbf{logarithm} (or the \textbf{natural logarithm}). It will be denoted by $L$, or by $\ln$.
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\end{definition}
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\begin{theorem}
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The logarithm is a strictly increasing function $L$ with domain $\{x \in \R : x > 0\}$ and range $\R$. The derivative of $L$ is given by
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\begin{enumerate}
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\item $L'(x)=1/x$ for $x >0$.The logarithm satisfies the functional equation
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\item $L(xy)=L(x)+L(y)$ for $x>0, y>0$. Moreover, we have
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\item $L(1)=0$ and $L(e)=1$,
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\item $L(x^r)=rL(x)$ for $x > 0$, $r \in \Q$,
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\item $\lim\limits_{x \to 0^+} L(x)=-\infty$ and $\lim\limits_{x \to \infty}L(x) = \infty$
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\end{enumerate}
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\end{theorem}
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\begin{definition}
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If $\alpha \in \R$ and $x > 0$, the number $x^\alpha$ is defined to be
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\[x^\alpha := e^{\alpha \ln x}=E(\alpha L(x))\]
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The function $x \mapsto x^\alpha$ for $x > 0$ is called the \textbf{power function} with exponent $\alpha$.
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\end{definition}
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\begin{theorem}
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If $\alpha \in \R$ and $x,y$ belong to $(0, \infty)$, then:
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\begin{enumerate}
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\item $1^\alpha = 1$
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\item $x^\alpha >0$
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\item $(xy)^\alpha = x^\alpha y^\alpha$
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\item $(x/y)^\alpha = x^\alpha / y^\alpha$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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If $\alpha, \beta \in \R$ and $x \in (0,\infty)$, then:
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\begin{enumerate}
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\item $x^{\alpha + \beta}=x^\alpha x^\beta$
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\item $(x^\alpha)^\beta = x^{\alpha \beta}=(x^\beta)^\alpha$
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\item $x^{-\alpha} = 1/x^\alpha$
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\item if $\alpha < \beta$, then $x^\alpha < x^\beta$ for $x > 1$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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Let $\alpha \in \R$. Then the function $x \mapsto x^\alpha$ on $(0,\infty)$ to $\R$ is continuous and differentiable and
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\[Dx^\alpha = \alpha x^{\alpha - 1}\ \ \text{for}\ \ x \in (0, \infty)\]
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\end{theorem}
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\begin{definition}
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Let $a>0,\ a \neq 1$. We define
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\[\log_a(x) := \frac{\ln(x)}{\ln(a)}\ \ \text{for}\ \ x \in (0,\infty)\]
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For $x \in (0,\infty)$, the number $\log_a(x)$ is called the \textbf{logarithm of $x$ to the base $a$}. The case $a=e$ yields the logarithm (or natural logarithm) function of \textit{Definition 8.3.1}. The case $a=10$ give sthe base 10 logarithm (or common logarithm) function $\log_{10}$ often used in computations. Properties of the functions $\log_a$ will be given in the exercises.
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\end{definition}
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\section{The Trigonometric Functions}
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\begin{theorem}
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There exist functions $C:\R \to \R$ and $S:\R\to\R$ such that
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\begin{enumerate}
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\item $C''(x)=-C(x)$ and $S''(x)=-S(x)$ for all $x \in \R$.
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\item $C(0)=1,\ C'(0)=0$, and $S(0)=0,\ S'(0)=1$.
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\end{enumerate}
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\end{theorem}
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\begin{corollary}
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If $C,\ S$ are the functions in \textit{Theorem 8.4.1}, then $C'(x)=-S(x)$ and $S'(x)=C(x)$ for all $x \in \R$. Moreover, these functions have derivatives of all orders.
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\end{corollary}
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\begin{corollary}
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The functions $C$ and $S$ satisfy the Pythagorean Identity:
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\[(C(x))^2+(S(x))^2=1\ \ \text{for}\ \ x \in \R\]
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\end{corollary}
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\begin{theorem}
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The functions $C$ and $S$ satisfying properties (1) and (2) of \textit{Theorem 8.4.1} are unique.
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\end{theorem}
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\begin{definition}
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The unique functions $C:\R\to\R$ and $S:\R\to\R$ such that $C''(x)=C(x)$ and $S''(x)=-S(x)$ for all $x \in \R$ and $C(0)=1,\ C'(0)=0$ and $S(0)=0,\ S'(0)=1$ are called the \textbf{cosine function} and the \textbf{sine function}, respectively. We ordinarily write
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\[\cos x := C(x)\ \ \text{and}\ \ \sin x := \S(x)\ \ \text{for}\ \ x \in \R\]
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\end{definition}
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\begin{theorem}
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If $f:\R\to\R$ is such that
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\[f''(x)=-f(x)\ \ \text{for}\ \ x \in \R\]
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then there exist real numbers $\alpha,\ \beta$ such that
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\[f(x)=\alpha C(x)+\beta S(x)\ \ \text{for}\ \ x \in \R\]
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\end{theorem}
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\begin{theorem}
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The function $C$ is even and $S$ is odd in the sense that
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\begin{enumerate}
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\item $C(-x)=C(x)$ and $S(-x)=-S(x)$ for $x \in \R$. If $x,\ y \in \R$, then we have the ``addition formulas".
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\item $C(x+y)=C(x)C(y)-S(x)S(y)$, $S(x+y)=S(x)C(y)+C(x)S(y)$
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\end{enumerate}
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\end{theorem}
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\begin{theorem}
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If $x \in \R,\ x \geq 0$, then we have
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\begin{enumerate}
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\item $-x \leq S(x) \leq x;$
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\item $1-\frac{1}{2}x^2 \leq C(x) \leq 1;$
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\item $x-\frac{1}{6}x^3 \leq S(x) \leq x;$
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\item $1-\frac{1}{2}x^2 \leq C(x) \leq 1-\frac{1}{2}x^2+\frac{1}{24}x^4$.
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\end{enumerate}
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\end{theorem}
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\begin{lemma}
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There exists a root $\gamma$ of the cosine function in the interval $(\sqrt{2}, \sqrt{3})$. Moreover $C(x) > 0$ for $x \in [0, \gamma)$. The number $2\gamma$ is the smallest positive root of $S$.
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\end{lemma}
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\begin{definition}
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Let $\pi:=2\gamma$ denote the smallest positive root of $S$.
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\end{definition}
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\begin{theorem}
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The functions $C$ and $S$ have period $2\pi$ in the sense that
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\begin{enumerate}
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\item $C(x+2\pi)=C(x)$ and $S(x+2\pi) = S(x)$ for $x \in \R$.
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\\Moreover we have
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\item $S(x)=C(\frac{1}{2}\pi - x) = -C(x+\frac{1}{2}\pi)$, $C(x)=S(\frac{1}{2}\pi-x)=S(x+\frac{1}{2}\pi)$ for all $x \in \R$.
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\end{enumerate}
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\end{theorem}
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