Asymptotic in LaTeX (\asymp)

Where it is used

Asymptotic-equivalence relation: two functions grow at the same rate up to constant factors.

In analysis and number theory f \asymp g means there are constants c, C > 0 with c g \le f \le C g for large arguments, equivalent to f = \Theta(g) in computer science. It is weaker than f \sim g, which requires the ratio to tend to 1. The glyph, a pair of opposed arcs, should not be confused with \approx or with \smile and \frown. It is a relation, so spacing is automatic. Some physics texts also use it loosely for asymptotically equal, so define it if precision matters.

Minimal compilable example

Paste this into an empty .tex file and compile it with pdfLaTeX, XeLaTeX or LuaLaTeX.

\documentclass{article}
\begin{document}
Asymptotic: $\asymp$

\[ \sum_{k=1}^{n} \frac{1}{k} \asymp \log n \]
\end{document}

Related symbols

• Defined As:=
• Succeeds or Equal\succeq
• Triangle Equal\triangleq
• Precedes or Equal\preceq
• Succeeds\succ
• Precedes\prec

Frequently asked questions

How do I write Asymptotic in LaTeX?

Type \asymp in math mode, for example $\sum_{k=1}^{n} \frac{1}{k} \asymp \log n$. It works in plain LaTeX without loading any package.

Which package do I need for \asymp?

None. \asymp is part of standard LaTeX, so it compiles with nothing in the preamble but the document class.

Can I use \asymp in text mode, or only in math mode?

Only in math mode. Outside $...$ LaTeX stops with "Missing $ inserted", so write $\asymp$ in running text. If you only need the glyph in text, paste the Unicode character ≍ instead (with XeLaTeX or LuaLaTeX and a font that has it).