\chapter{Advanced integration techniques}

\section{Series method}

At this point we have at our disposal the fact that uniform convergence permits the swapping of limits and integrals and the BCT by Arzela. If we can find series representations of the functions (say from applying Taylor's theorem) we wish to integrate and one of these theorems permits the swapping of limits and integrals, then we can integrate over the series instead.







\section{Leibniz integral rule}

There is a particular theorem regarding integration that 

\begin{theorem}[Leibniz integral rule]

\[F(t) = \int^{u_1 (t)}_{u_2 (t)} f(x,t)dx \]
\[F'(t) = f(b(t),t)b'(t) - f(a(t),t)a'(t) + \int^{u_1 (t)}_{u_2 (t)} f_{t}(x,t)dx  \]
\end{theorem}

Like many proofs, this is dicking around with integrals in an obvious way, 2 applications of the integral MVT, and one application of BCT.

We can use this technique to evaluate integral of a $\mathbb{R}$ continuous function $f$ by introducing some $\mathbb{R}^2$ function $g$, where there is some constant $c$ that recovers $f$ back from $g$ (so we have $g(x,c)=f(x)$). This turns out to be quite the powerful trick.

\section{The Gamma and Beta function}

Another method for solving integrals is reducing them to a case of the \emph{Gamma function}; a function that is just generally awesome.

\[\Gamma (z) = \int^{\infty}_{0} t^{z-1} e^{-t}dt \]

The reason that it is desirable to find a Gamma function representation is because the Gamma function obeys many useful formulae for calculation



\[\Gamma (z+1) = z \Gamma (z) \]
\[\Gamma (n) = (n-1)! \]


\[\Gamma (\frac{1}{2}) = \sqrt{\pi} \]

Reflection
\[\Gamma (1-z) \Gamma (z) = \frac{\pi}{\sin (\pi z)} \]

Duplication
\[\Gamma (z) \Gamma (z+\frac{1}{2}) = 2^{1-2z}\sqrt{\pi} \Gamma (2z) \]