Real Analysis
Exchange

Last Updated: November, 2004

\documentclass{article}

\usepackage{amssymb,amsmath}

\begin{document}

\section{Long Equations}

Here is a long equation without line number.

\begin{align*}
\int_{G} \Theta(f_{\underline{\varepsilon}} (t)) &\,d \mu (t) =  -\int _{G}
f_{\underline{\varepsilon}}^{4}(t)\,d \mu (t) \\
&+(b^{2} +
2a^{2}) \int _{G} f_{\underline{\varepsilon}}^{2}(t)\, d \mu (t) +2ab^{2}\int _{G} f_{\underline{\varepsilon}}(t)\, d \mu (t)+
a^{2}b^{2} - a^{4}.
\end{align*}

Here is the same long equation with a single equation number. The "notag" is used to suppress numbering the first line.

\begin{align}\label{long1}
\int_{G} \Theta(f_{\underline{\varepsilon}} (t)) &\,d \mu (t) =  -\int _{G}
f_{\underline{\varepsilon}}^{4}(t)\,d \mu (t) \notag\\
&+(b^{2} +
2a^{2}) \int _{G} f_{\underline{\varepsilon}}^{2}(t)\, d \mu (t) +2ab^{2}\int _{G} f_{\underline{\varepsilon}}(t)\, d \mu (t)+
a^{2}b^{2} - a^{4}.
\end{align}

Here is the same long equation with a single equation number, but centered. Here the combination of "equation" and "split" environments is used. We prefer this version for numbered long equations.

\label{long2}
\begin{split}
\int_{G} \Theta(f_{\underline{\varepsilon}} (t)) &\,d \mu (t) =  -\int _{G}
f_{\underline{\varepsilon}}^{4}(t)\,d \mu (t) \\
&+(b^{2} +
2a^{2}) \int _{G} f_{\underline{\varepsilon}}^{2}(t)\, d \mu (t) +2ab^{2}\int _{G} f_{\underline{\varepsilon}}(t)\, d \mu (t)+
a^{2}b^{2} - a^{4}.
\end{split}

\section{Multiline Equations}

\begin{enumerate}
\item[]  {\large\sc Example 1.}

Here is a string of equations typeset with gather.

\begin{gather*}
a_n=\frac{1}{ 2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\;dx
=\frac{1}{
2\pi}\int_{-\pi}^{\pi}f_0(x)e^{-inx}\;dx \\
=\lim_{n\to\infty}\frac{1}{
2\pi}\int_{-\pi}^{\pi}f_{\delta_n}(x)e^{-inx}\;dx
=\lim_{n\to\infty}\frac{1}{ 2\pi}\int_{-\infty}^{\infty}g_{\delta_n}(t)
\left(\int_{-\pi}^{\pi}\frac{e^{-inx}}{
|x-t|^{1-\alpha}}\;dx\right)dt \\
=\frac{1}{
2\pi}\int_{-\pi}^{\pi}U_{\alpha}^{\mu}(x)e^{-inx}\;dx.
\end{gather*}

Here is the same set of equations typeset with align.

\begin{align*}
a_n&=\frac{1}{ 2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\;dx
=\frac{1}{
2\pi}\int_{-\pi}^{\pi}f_0(x)e^{-inx}\;dx \\
&=\lim_{n\to\infty}\frac{1}{
2\pi}\int_{-\pi}^{\pi}f_{\delta_n}(x)e^{-inx}\;dx
=\lim_{n\to\infty}\frac{1}{ 2\pi}\int_{-\infty}^{\infty}g_{\delta_n}(t)
\left(\int_{-\pi}^{\pi}\frac{e^{-inx}}{
|x-t|^{1-\alpha}}\;dx\right)dt \\
&=\frac{1}{
2\pi}\int_{-\pi}^{\pi}U_{\alpha}^{\mu}(x)e^{-inx}\;dx.
\end{align*}

Each of these two versions has advantages over the other. The first looks better on the page while the second highlights the fact that a formula for $a_n$ is underway.

The second formulation introduces an "overfill" on the second line, however, so must be altered to fit on the page properly. Here is a reformulation.

\begin{align*}
a_n&=\frac{1}{ 2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\;dx
=\frac{1}{ 2\pi}\int_{-\pi}^{\pi}f_0(x)e^{-inx}\;dx \\
&=\lim_{n\to\infty}\frac{1}{
2\pi}\int_{-\pi}^{\pi}f_{\delta_n}(x)e^{-inx}\;dx \\
&=\lim_{n\to\infty}\frac{1}{ 2\pi}\int_{-\infty}^{\infty}g_{\delta_n}(t)
\left(\int_{-\pi}^{\pi}\frac{e^{-inx}}{
|x-t|^{1-\alpha}}\;dx\right)dt \\
&=\frac{1}{
2\pi}\int_{-\pi}^{\pi}U_{\alpha}^{\mu}(x)e^{-inx}\;dx.
\end{align*}

We prefer this third version under most circumstances, but there are certainly times when one of the other two works best.

\item[] {\large\sc Example 2.}

Here is a common alignment first without line number, then with.

\begin{align*}
P(G_N)&\leq\sum_{\frac{\mu}{2}<j\leq\mu}P\left(\|B'_j\|_T>\frac{q_j}{4}\frac{N}{T}\right)\\
&\leq\sum_{\frac{\mu}{2}<j\leq\mu}\frac{4T}{Nq_j}C'_Tq_j\left(\log
p_{j+1}\right)^{3/2}\\
&\leq\frac{4T}{N}C'_T\mu\left(\log p_{\mu+1}\right)^{3/2}\\
&\leq\frac{4TC'_T}{N}\mu\exp\left(\frac{3(\mu+1)l}{\gamma}\right),
\end{align*}

\label{eq1}
\begin{split}
P(G_N)&\leq\sum_{\frac{\mu}{2}<j\leq\mu}P\left(\|B'_j\|_T>\frac{q_j}{4}\frac{N}{T}\right)\\
&\leq\sum_{\frac{\mu}{2}<j\leq\mu}\frac{4T}{Nq_j}C'_Tq_j\left(\log
p_{j+1}\right)^{3/2}\\
&\leq\frac{4T}{N}C'_T\mu\left(\log p_{\mu+1}\right)^{3/2}\\
&\leq\frac{4TC'_T}{N}\mu\exp\left(\frac{3(\mu+1)l}{\gamma}\right),
\end{split}

Here is one more example of a numbered multiline equation.

\label{EQ29}
\begin{split}
P_k(s)=&\frac{1}{h_k}+\frac{e^{-s}}{h_k-1}+\dots+\frac{e^{-(h_k-1)s}}{1}-\frac{e^{-(h_k+1)s}}{1}-\dots-\frac{e^{-2h_ks}}{h_k}\\
=&
\sum_{j=0}^{h_k-1}\frac{e^{-js}}{h_k-j}-\sum_{j=0}^{h_k-1}\frac{e^{-(2h_k-j)s}}{h_k-j}\\
=&\sum_{n=1}^{h_k}\frac{e^{-(h_k-n)s}-e^{-(h_k+n)s}}{n}.
\end{split}

\end{enumerate}

\end{document}