Authorea

Displayed equations

\label{DispEqSection}

When an equation is important and/or tall and/or complicated, you should display it on a line by itself, with a number. To do this, you put \begin{equation} before the equation and \end{equation} after it, as in

\begin{equation} \int_{0}^{\infty}\!\frac{x^{3}}{e^{x}-1}\,dx=6\sum_{k=1}^{\infty}\frac{1}{k^{4}}=6\left(\frac{\pi^{4}}{90}\right)=\frac{\pi^{4}}{15}.\\ \end{equation}

This example also shows how to make the sum and integral symbols, big parentheses, and built-up fractions. (Don’t put built-up fractions in a non-displayed equation, because there won’t be enough vertical space in AJP’s final, single-spaced paragraphs. Use the slashed form, \(x^{3}/(e^{x}-1)\), instead.)

If you want to refer to an equation elsewhere in your manuscript, you can give it a label. For example, in the equation

\begin{equation} \label{deriv} \label{deriv}\frac{\Delta x}{\Delta t}\mathop{\longrightarrow}_{\Delta t\rightarrow 0}\frac{dx}{dt}=\lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}\\ \end{equation}

we’ve inserted \label{deriv} to label this equation deriv.\cite{labelnames} To refer to Eq. (\ref{deriv}), we then type \ref{deriv} .\cite{footnotes} Notice that AJP’s style conventions also require you to put the equation number in parentheses when you refer to it, and to abbreviate “Eq.” unless it’s at the beginning of a sentence.

Some equations require more complicated layouts. In the equation

\begin{equation} E_{n}=(n+\tfrac{1}{2})\hbar,\quad\textrm{where}\ n=0,1,2,\ldots,\\ \end{equation}

we’ve used \quad to leave a wide space and \textrm to put “where” in plain Roman type. To create a matrix or column vector, as in

\begin{equation} \begin{bmatrix}t^{\prime}\\ x^{\prime}\\ \end{bmatrix}=\begin{pmatrix}\gamma&-\beta\gamma\\ -\beta\gamma&\gamma\\ \end{pmatrix}\begin{bmatrix}t\\ x\\ \end{bmatrix},\\ \end{equation}

you can use the pmatrix and/or bmatrix environment, for matrices delimited by parentheses and/or brackets. There’s also a plain matrix environment that omits the delimiters. In this and other examples of LaTeX tables and arrays, the & character serves as a “tab” to separate columns, while the \\ control sequence marks the end of a row.

For a list of related equations, with nicely lined-up equals signs, use the eqnarray environment:

\begin{aligned} \oint\vec{B}\cdot d\vec{\ell} & = & -\frac{d\Phi_{E}}{dt}; \\ \oint\vec{E}\cdot d\vec{\ell} & = & \mu_{0}\epsilon_{0}\frac{d\Phi_{B}}{dt}+\mu_{0}I.\\ \end{aligned}

You can also use eqnarray to make a multi-line equation, for example,

\begin{aligned} \mathcal{Z} & = & 1+e^{-(\epsilon-\mu)/kT}+e^{-2(\epsilon-\mu)/kT}+\cdots \\ & = & 1+e^{-(\epsilon-\mu)/kT}+(e^{-(\epsilon-\mu)/kT})^{2}+\cdots\nonumber \\ & = & \frac{1}{1-e^{-(\epsilon-\mu)/kT}}.\nonumber \\ \end{aligned}

Here the first column of the second and third lines is empty. Note that you can use \nonumber within any line to suppress the generation of an equation number; just be sure that each multi-line equation has at least one number.

Another commonly used structure is the cases environment, as in

\begin{equation} m(T)=\begin{cases}0&T>T_{c}\,,\\ \bigl{(}1-[\sinh 2\beta J]^{-4}\bigr{)}^{1/8}&T<T_{c}\,.\end{cases}\\ \end{equation}

At AJP we require that you put correct punctuation before and after every displayed equation, treating each equation as part of a correctly punctuated English sentence.\cite{mermin} The preceding examples illustrate good equation punctuation.