Splitting an equation over multiple lines in LaTeX is a common requirement for dealing with long mathematical expressions.
There are several environments you can use. Among them, the most flexible and widely used are multline and split environments, from the amsmath package.
Break points and alignment points for split equation
Before jumping directly to main point, we will understand the importance of two concepts.
Significance of \\ and \\[length]
The double backslash \\ and its variant \\[length] are used to manage line breaks and vertical spacing in various environments.
And,[length] argument specifies how much extra space is added.
Significance of & symbol for creating alignment points
To split an equation over two or more lines in LaTeX and ensure that the lines are aligned at a specific point.
The & symbol plays a crucial role in this environment, indicating the alignment point for each line of equation.
Multline environment without align
Breaking a long equation into multiple lines in LaTeX is efficiently handled by multline environment, provided by the amsmath package.
Alignment of the Breaking equation is not possible with this environment.
\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage[top=1cm,buttom=1cm]{geometry} % for margin
\begin{document}
\section*{Taylor Expansion}
Here, the Taylor series expansion is broken into two lines, showcasing how to handle series expansions in multi line format.
\begin{multline}
e^{\sin x} = 1 + \sin x + \frac{\sin^2 x}{2!} + \frac{\sin^3 x}{3!} +
\frac{\sin^4 x}{4!} + \frac{\sin^5 x}{5!} + \\ \frac{\sin^6 x}{6!} + \frac{\sin^7 x}{7!} + \frac{\sin^8 x}{8!} +\frac{\sin^9 x}{9!}\cdots
\end{multline}
\section*{Long integral}
This example shows a long integral expression broken into two lines. The integral and its limits are on the first line, and the series expansion follows on the second.
\begin{multline}
\int_0^\infty e^{-x^2} dx = 1 + 2x - 3x^3 + 4x^4 - 5x^5 + 6x^6 - \\ 7x^7 + 8x^8 -9x^9 +10x^{10}-11x^{11}+\cdots
\end{multline}
\section*{Equation with Functions}
Here, a function consisting of various trigonometric functions is broken into two lines for clarity.
\begin{multline}
f(x) = \sin(x) + \cos(x) + \tan(x) + \cot(x) + \\
\sec(x) + \csc(x) + \arcsin(x) + \arccos(x) + \arctan(x)
\end{multline}
\section*{Derivative of a Trigonometric Function}
This example demonstrates the derivative of a combination of trigonometric functions, neatly divided into two lines.
\begin{multline}
\frac{d}{dx} \left( \sin(x^2) + \cos^2(x) - \tan^{-1}(x) +\ln(\sin\,x)\right) = \\
2x \cos(x^2) - 2\sin(x)\cos(x) - \frac{1}{1+x^2} + \frac{\cos x}{\sin x}
\end{multline}
This equation includes a variety of complex calculus elements.
\begin{multline}
\int_{0}^{\infty} \left[ \frac{\sin(x)}{x} - e^{-x^2} \right] dx = \lim_{y \to 0} \left( \frac{1}{y} \int_{0}^{y} \log(1 + x^2) dx \right) - \\
\sum_{n=1}^{\infty} \frac{(-1)^n}{n^3} + \int_{-\pi}^{\pi} e^{x \cos \theta} \cos(x \sin \theta) d\theta + \\
\frac{d^2}{dx^2} \left( \frac{x^4 - 6x^2 + 8x - 3}{x^2 + 1} \right) -\iint_{D} e^{-(x^2 + y^2)} \, dx \, dy
\end{multline}
\end{document}
Output :
And using \\ symbol to indicate the end of current line and the start of next line. And number equation will return with the last line.
Use split environment inside equation environment
The equation environment returns an automatic number equation. When you use split environment inside it, the entire block of equations is treated as one single entity, receiving one equation number.
And split environment allows you to align the split parts of equation at any desired place. Which is fixed by & symbol.
\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage[margin=1.5cm]{geometry}
\begin{document}
\section*{Long Integral Equation}
A long integral expression is broken into two lines, with the continuation indented for clarity.
\begin{equation}
\begin{split}
\int_0^\infty e^{-x^2} dx = & 1 + 2x - 3x^3 + 4x^4 - \\
& 5x^5 + 6x^6 - 7x^7 + \cdots
\end{split}
\end{equation}
\section*{Equation with Functions and Multiple Alignments}
The function is split into three lines, each part aligned at the equal sign for consistency.
\begin{equation}
\begin{split}
f(x) = & \sin(x) + \cos(x) + \tan(x) + \\[5pt]
& \cot(x) + \sec(x) + \csc(x) + \\[5pt]
& \arcsin(x) + \arccos(x) + \arctan(x)
\end{split}
\end{equation}
\section*{Summation and Product}
This complex equation combines summation and product symbols, breaking down the summation into its components and then showing its relationship to a product.
\begin{equation}
\begin{split}
S_n = & \sum_{i=1}^{n} \left( a_i + b_i \right) \\
= & a_1 + b_1 + a_2 + b_2 + a_3 + b_3 + \cdots + \\
& a_{n-1} + b_{n-1} + a_n + b_n \\
= & \prod_{j=1}^{n} c_j + \sum_{k=1}^{n} d_k
\end{split}
\end{equation}
\section*{Nested Fractions and Functions}
This equation features nested fractions and trigonometric functions, carefully broken into two lines for clarity.
\begin{equation}
\begin{split}
f(x) = \frac{1}{2} \left[ \frac{3x^2 - 2x + 1}{x^3 - x + 4} + \right. & \\
& \left. \frac{\sin(x) - \cos(x)}{\sqrt{x^2 + 1}} \right]
\end{split}
\end{equation}
\section*{Integral with Limits and Series Expansion}
This equation shows an integral with its limits and its corresponding series expansion, broken into two lines for detailed explanation.
\begin{equation}
\begin{split}
\int_{a}^{b} e^{x^2} dx = & \left. \frac{e^{x^2}}{2x} \right|_a^b - \int_{a}^{b} x e^{x^2} dx \\[5pt]
= & \sum_{n=0}^{\infty} \frac{(b^{2n+1} - a^{2n+1})}{n!(2n+1)}
\end{split}
\end{equation}
\end{document}Output :
Auto break equation by breqn package
The dmath environment from breqn package in LaTeX is designed to automatically break long equations into multiple lines at appropriate places. This package takes care of line breaks and alignment automatically.
Enclose your long equation within \begin{dmath} and \end{dmath}.
\documentclass[11pt]{article}
\usepackage{breqn,lipsum}
\usepackage[top=1.5cm]{geometry}
\begin{document}
\section*{Without space}
\lipsum[5][1-4]
\begin{dmath}
e^x = 1 + (x+1) + \frac{x^2 + 2}{2!} + \frac{x^3+3}{3!} + \frac{x^4+4}{4!} + \frac{x^5+5}{5!} + \frac{x^6+6}{6!} + \frac{x^7+7}{7!} + \frac{x^8+8}{8!} + \frac{x^9+9}{9!} + \frac{x^{10}+10}{10!}\cdots
\end{dmath}
\section*{With space}
\lipsum[4][1-4]
\begin{dmath}[spread=10pt]
e^x = 1 + (x+1) + \frac{x^2 + 2}{2!} + \frac{x^3+3}{3!} + \frac{x^4+4}{4!} + \frac{x^5+5}{5!} + \frac{x^6+6}{6!} + \frac{x^7+7}{7!} + \frac{x^8+8}{8!} + \frac{x^9+9}{9!} + \frac{x^{10}+10}{10!}\cdots
\end{dmath}
\lipsum[3][1-4]
\begin{dmath}[spread=5pt]
\sin(x) + \cos(x^2) + \tan(x^3) + \cot(x^4) + \sec(x^5) + \csc(x^6) + \arcsin(x^7) + \arccos(x^8) + \arctan(x^9)
\end{dmath}
\lipsum[2][1-3]
\begin{dmath}[spread=8pt]
\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} + \frac{x^9}{9} - \frac{x^{10}}{10} + \frac{x^{11}}{11} - \frac{x^{12}}{12} + \frac{x^{13}}{13} - \frac{x^{14}}{14} \cdots
\end{dmath}
\lipsum[8][1-4]
\begin{dmath}
\lim_{x \to 0} \left( \frac{e^{x} - e^{-x}}{x} + \frac{\tan(x)}{x^2} - \frac{\arcsin(x)}{x^3} + \frac{\sqrt[3]{x^4 + 1} - 1}{x} + \frac{\sin 2x}{x} + \frac{\tan 2x}{x} + \frac{\sin 3x}{x} + \frac{\tan 3x}{x} \right )
\end{dmath}
\end{document}Output :
In this case, the process of adding vertical space is completely different. This package contains a per-build [spread=length] option, and can set the length in length argument.
Use align environment
With the align environment you can break long equations, but the align environment will provide numbered equations for each line.
For a long equation, there should be a number equation. So, it doesn’t matter how many parts are divided.
For this, the nonumber command is used to control the number equation.
\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage[top=1cm]{geometry}
\begin{document}
\section*{Without nonumber command(not best method)}
Here, the derivative of a complex function is calculated. The equation is broken at logical points, with each term of the derivative aligned on separate lines.
\begin{align}
\frac{d}{dx} \left( x^4 \sin(x^2) - \frac{1}{x^2 + 1} + \ln(x) \right) = \; & 4x^3 \sin(x^2) + 2x^5 \cos(x^2) + \\ & \frac{2x}{(x^2 + 1)^2} + \frac{1}{x}
\end{align}
\section*{With nonumber command}
Here, the derivative of a complex function is calculated. The equation is broken at logical points, with each term of the derivative aligned on separate lines.
\begin{align}
\frac{d}{dx} \left( x^4 \sin(x^2) - \frac{1}{x^2 + 1} + \ln(x) \right) = \; & 4x^3 \sin(x^2) + 2x^5 \cos(x^2) +\nonumber \\ & \frac{2x}{(x^2 + 1)^2} + \frac{1}{x}
\end{align}
\section*{Multivariable Integral}
This example demonstrates a multivariable integral over a domain DD. The integral is split into two parts, each computed over different intervals,
\begin{align}
\int\int_D (x^2 + y^2) \,dx\,dy = & \int_0^1 \int_0^{\sqrt{1-y^2}} (x^2 + y^2) \,dx\,dy \,+ \nonumber \\[4pt]
& \int_1^2 \int_0^{\sqrt{4-y^2}} (x^2 + y^2) \,dx\,dy
\end{align}
\section*{Advanced Derivative with Trigonometric Functions}
Here, a derivative of a function involving both polynomial and trigonometric terms.
\begin{align}
f(x) = & \sin(x) + \cos(x) + \tan(x) + \nonumber \\
& \sec(x) + \csc(x) + \arcsin(x) + \arccos(x) + \arctan(x)
\end{align}
\begin{align}
\frac{d}{dx} \left( x^3 \cos(x^2) + \frac{\tan(x)}{x} \right) &= 3x^2\cos(x^2) - 2x^4\sin(x^2) + \nonumber \\ &\quad \frac{\sec^2(x)}{x} - \frac{\tan(x)}{x^2}
\end{align}
\end{document}Output :
Aligned with equation environment
Our last method is aligned, which we will use within the equation environment. aligned environment is completely similar to split environment.
And, you can maintain the alignment and spacing nicely. Look at the code below.
\documentclass[11pt]{article}
\usepackage{amsmath,lipsum}
\usepackage[margin=1.5cm]{geometry}
\begin{document}
\begin{equation}
\begin{aligned}
\ln(1+x) = \, & x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \frac{x^6}{6} + \frac{x^7}{7} - \frac{x^8}{8} +\\[5pt] & \frac{x^9}{9} - \frac{x^{10}}{10} + \frac{x^{11}}{11} - \frac{x^{12}}{12} + \frac{x^{13}}{13} - \frac{x^{14}}{14} \cdots
\end{aligned}
\end{equation}
\lipsum[4][3-6]
\begin{equation}
\begin{aligned}
f(x) = & \sin(x) + \cos(x) + \tan(x) + \sec(x) + \\
& \csc(x) + \arcsin(x) + \arccos(x) + \arctan(x)
\end{aligned}
\end{equation}
\lipsum[3][3-6]
\begin{equation}
\begin{aligned}
f(x) = e^x \sin(x) + & e^{-x} \cos(x) - \tan^2(x) + \\
& \sqrt{x} \ln(x) - \arctan(x) + \frac{1}{1 + x^2}
\end{aligned}
\end{equation}
\lipsum[7][3-8]
\begin{equation}
\begin{aligned}
f(x) = \frac{1}{2} \left[ \frac{3x^2 - 2x + 1}{x^3 - x + 4} + \right. & \\
& \left. \frac{\sin(x) - \cos(x)}{\sqrt{x^2 + 1}} \right]
\end{aligned}
\end{equation}
\end{document}Output :
Best practice
Break point: Break equations at natural points, like plus signs, minus signs, or equal signs. Avoid breaking an equation in the middle of a complex term or function.
Align point: Use & to align the equations at the desired point. It’s typically used before an equal sign or an operation sign.
Vertical space: Where vertical space is required, vertical space needs to be provided. As a result, mathematical documents look more beautiful.
Conclusion
As many methods as possible, all methods or environments have solved this problem. Split environment is best if you want alignment.
And if you don’t have the requirement of alignment, then you can use multline environment or auto breaking method.
