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\begin{center}
{\large Math tools }
\vspace{3 mm}
\end{center}

Here are some useful theorems and definitions from advanced calculus and real
analysis. Most of them are in the books on reserve in the math library.

\paragraph{Leibnitz's Rule} Let $f(x,\theta)$ and $\frac{\partial f}{\partial
\theta}$ be continuous in some region of the $(x,\theta)$ plane including $u_1
\leq x \leq u_2$ and $a \leq \theta \leq b$, and let $u_1(\theta)$ and
$u_2(\theta)$ have continuous derivatives for $a \leq \theta \leq b$. Then

\begin{displaymath}
            \frac{d}{d \theta} \int_{u_1(\theta)}^{u_2(\theta)} f(x,\theta) \, dx
    =       \int_{u_1(\theta)}^{u_2(\theta)} 
                \frac{\partial}{\partial \theta}f(x,\theta) \, dx 
    \;+\;   f(u_2(\theta),\theta) \frac{d}{d\theta} u_2(\theta)
    \;-\;   f(u_1(\theta),\theta) \frac{d}{d\theta} u_1(\theta)
\end{displaymath}
for $a \leq \theta \leq b$.

\paragraph{Taylor's Theorem} Let the $n$th derivative $f^{(n)}$ be continuous
in $[a,b]$ and differentiable in $(a,b)$, with $x$ and $x_0$ in $(a,b)$. Then
there exists a point $\xi$ between $x$ and $x_0$ such that
\begin{eqnarray*}
    f(x) & = & f(x_0) \;+\; f^\prime(x_0)\,(x-x_0) 
                \;+\; \frac{f^{\prime\prime}(x_0)(x-x_0)^2}{2!}
                \;+\; \ldots 
                \;+\; \frac{f^{(n)}(x_0)(x-x_0)^n}{n!} \\
         & + &     \; \frac{f^{(n+1)}(\xi)(x-x_0)^{n+1}}{(n+1)!}
\end{eqnarray*}
where $R_n = \frac{f^{(n+1)}(\xi)(x-x_0)^{n+1}}{(n+1)!}$ is called the
\emph{remainder term}. If $R_n \rightarrow 0$ as $n \rightarrow \infty$, the
resulting infinite series is called the \emph{Taylor Series} for $f(x)$. There
are other forms for the remainder term (with a different value of $\xi$) that
sometimes prove useful.

\paragraph{Fubini's Theorem} If a double integral (sum) converges absolutely,
then the order of integration (summation) may be exchanged. If the quantity
being integrated (or added up) is positive, then integration (summation) may
always be exchanged, and if the result is $\infty$ in one direction, it is
$\infty$ in the other direction too. See Fraser's \emph{Probability and
Statistics} or a measure theory book like Royden's \emph{Real Analysis} for
more precision.

\paragraph{Uniform Convergence of Sums} Let $u_1(x),\, u_2(x), \, \ldots \,$
be a sequence of functions, and define $S_n(x) = \sum_{k=1}^n
u_k(x)$. $\lim_{n \rightarrow \infty} S_n(s) = S(x)$ means $\forall \,
\epsilon>0, \, \exists \, N \ni$ if $n>N$, then $|S_n(x)-S(x)|<\epsilon$. In
general, $N$ will depend on $x$ as well as $\epsilon$. If $N$ only depends on
$\epsilon$ (often for all $x$ in some interval) then we will say that $S_n(x)$
converges \emph{uniformly} to $S(x)$ (again, often for the $x$ in that
interval).

\paragraph{Tests for Uniform Convergence of Sums}
\begin{description}
   \item[Weierstrass M test] If there is a sequence of positive constants
$M_1,\, M_2, \, \ldots \,$ such that $|u_n(x)| \leq M_n$ in some interval for
each $n$, and if $\sum_{n=1}^\infty M_n$ converges, then $\sum_{n=1}^\infty
u_n(x)$ converges uniformly in that interval.
   \item[Dirichlet's test] If the sequence of constants $a_n\downarrow 0$, and
there exist constants $N$ and $P$ such that for $a \leq x \leq b$,
$|S_n(x)|<P\;\forall\,n>N$, then the series $\sum_{n=1}^\infty a_nu_n(x)$ is
uniformly convergent for $a \leq x \leq b$.  \end{description}

\paragraph{Theorems about Uniform Convergence of Sums} 
\begin{itemize}
   \item If $u_n(x)$ is continuous in $[a,b]$ for $n=1, 2, \ldots$ , and if
$S_n(x)$ converges uniformly to $S(x)$ in $[a,b]$, then $\lim_{x \rightarrow
x_0} \sum_{n=1}^\infty u_n(x) = \sum_{n=1}^\infty \lim_{x \rightarrow x_0}
u_n(x) = \sum_{n=1}^\infty u_n(x_0)$, where $x_0 \in [a,b]$, and right or left
hand limits are used if $x_0$ is an endpoint of $[a,b]$.
   \item If $u_n(x)$ has a continuous derivative in $[a,b]$ for $n=1, 2,
\ldots$ , and if $S_n(x)$ converges to $S(x)$ while $\sum_{n=1}^\infty
\frac{d}{dx} u_n(x)$ converges uniformly in $[a,b]$, then $\frac{d}{dx}
\sum_{n=1}^\infty u_n(x) = \sum_{n=1}^\infty \frac{d}{dx} u_n(x)$.
\end{itemize}

\paragraph{Power Series} A series of the form $ \sum_{n=0}^\infty a_n (x-a)^n$
is called a \emph{power series} in $x$. If it converges for $|x|<R$ and
diverges for $|x|>R$, the constant $R$ is called the \emph{radius of
convergence}.

A power series converges absolutely and uniformly in any interval that lies
entirely within its interval of convergence, and hence may be differentiated
term by term there. It may also be integrated term by term over any interval
strictly within the interval of convergence.

\pagebreak

\paragraph{Uniform Convergence of Integrals} The integral $\int_a^\infty
f(x,\theta)\,dx$ is said to converge uniformly for $\theta \in
[\theta_1,\theta_2]$ if $\forall \, \epsilon>0, \, \exists \, N \in \mathbb{R}
\ni$ if $u>N$, then $|\int_a^\infty f(x,\theta)\,dx-\int_a^u
f(x,\theta)\,dx|<\epsilon$, where $N$ depends on $\epsilon$ but not $\theta$.

\paragraph{Tests for Uniform Convergence of Integrals}
\begin{description}
   \item[Weierstrass M test] If there is a function $M(x) \geq 0$ such that
$|f(x,\theta)| \leq M(x)$ for $\theta \in [\theta_1,\theta_2]$ and $x>a$, and
if $\int_a^\infty f(x,\theta)\,dx$ converges, then $\int_a^\infty
f(x,\theta)\,dx$ is uniformly and absolutely convergent for $\theta$ in
$[\theta_1,\theta_2]$.
   \item[Dirichlet's test] If the sequence of functions $\psi(x)\downarrow 0$
as $x \rightarrow \infty$, and there exists a constant $P$ such that
$|\int_a^u f(x,\theta)\,dx|<P$ for all $u>a$ and $\theta \in
[\theta_1,\theta_2]$, then $\int_a^\infty f(x,\theta)\psi(x)\,dx$ is uniformly
convergent for $\theta \in [\theta_1,\theta_2]$.  \end{description}

\paragraph{Theorems about Uniform Convergence of Integrals} 
\begin{itemize}
   \item If $f(x,\theta)$ is continuous for $x \geq a$ and $\theta \in
[\theta_1,\theta_2]$, and if $\int_a^\infty f(x,\theta)\,dx$ converges
uniformly for $\theta \in [\theta_1,\theta_2]$, then   
$\lim_{\theta \rightarrow \theta_0} \int_a^\infty f(x,\theta)\,dx =
\int_a^\infty \lim_{\theta \rightarrow \theta_0} f(x,\theta)\,dx =
\int_a^\infty f(x,\theta_0)\,dx$, where $\theta_0 \in [\theta_1,\theta_2]$,
and right or left hand limits are used if $\theta_0$ is an endpoint of
$[a,b]$.
   \item If $f(x,\theta)$ has a continuous derivative with respect to $\theta$
in $[\theta_1,\theta_2]$ for $x \geq a$ and if $\int_a^u f(x,\theta)\,dx$
converges to $\int_a^\infty f(x,\theta)\,dx$ while $\int_a^\infty
\frac{\partial}{\partial \theta}f(x,\theta)\,dx$ converges uniformly in
$[\theta_1,\theta_2]$, then $\frac{\partial}{\partial \theta}\int_a^\infty
f(x,\theta)\,dx = \int_a^\infty \frac{\partial}{\partial
\theta}f(x,\theta)\,dx$, provided $a$ does not depend on $\theta$. If it does,
use Leibnitz's rule.

\end{itemize}

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