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Write the Laplace transform in LaTeX with \mathcal{L} for the operator, \int_{0}^{\infty} for the integral and ^{-1} for the inverse. The complete definition is \mathcal{L}\{f(t)\}(s) = \int_{0}^{\infty} f(t)\,e^{-st}\,dt, and it compiles in a standard LaTeX document or an Overleaf project without extra packages for the basic form. This guide gives the copy-paste source, a reusable macro, the inverse transform, the alternative script letters, and the errors that most often break the output.
What it is used for in research
The Laplace transform appears in manuscripts, theses and technical reports in control engineering, signal processing, differential-equation modelling and pharmacokinetic or physical modelling. LaTeX has no dedicated Laplace command: the convention is to set the operator as a calligraphic capital L (\mathcal{L}) and build the definition from the ordinary integral syntax. The sections below show each piece so you can paste and adapt it.
Install or access (official source only)
No Laplace-specific package is needed for the examples in steps 1 to 4. You need a working LaTeX environment: either an online editor such as Overleaf, or a local TeX distribution. Two optional packages are mentioned later and both are distributed through CTAN, the Comprehensive TeX Archive Network:
- amsmath: needed for
\DeclareMathOperatorand for multiple-integral commands (Overleaf documentation, read 2026-11-08). - amssymb: the Overleaf mathematical-fonts page states that some math font styles require
\usepackage{amssymb}in the preamble. - mathrsfs: provides
\mathscr(see the CTAN mathrsfs page, read 2026-11-08).
Step-by-step task
1. Typeset the operator symbol (the “latex laplace symbol”)
Use \mathcal{L} inside math mode. It prints a calligraphic L. The Overleaf mathematical-fonts page documents \mathcal as the command for calligraphic letters.
$\mathcal{L}$2. Write the transform of a function
Wrap the function in escaped braces so they print: \{ and \}.
\[ \mathcal{L}\{f(t)\}(s) \]3. Write the defining integral
The Overleaf integrals page gives the pattern \int_{lower}^{upper}, with an example \int_{a}^{b} x^2 \,dx, where \, inserts a thin space before the differential. \infty is the infinity symbol listed under the miscellaneous symbols on the Overleaf symbol list. Combined:
\[
\mathcal{L}\{f(t)\}(s) = \int_{0}^{\infty} f(t)\,e^{-st}\,dt
\]Use the display form (\[ ... \]) for this formula. The Overleaf page notes that inline and display math place the integral limits differently, so a long integral reads better displayed.
4. Add the inverse transform
The inverse is conventionally written with a superscript minus one on the same operator:
\[ f(t) = \mathcal{L}^{-1}\{F(s)\}(t) \]5. Make it reusable
If you use the operator many times, define a macro once in the preamble with the standard \newcommand, so a later change of symbol is made in one place:
\newcommand{\Lap}{\mathcal{L}}
...
\[ \Lap\{f(t)\}(s) \]Alternatively, \DeclareMathOperator{\name}{text} from amsmath defines an operator that prints in roman type; the Overleaf operators page shows the form \DeclareMathOperator{\arctg}{arctg} in the preamble. That suits a named operator, not the single calligraphic L.
Output and how to read it
Step 3 prints a calligraphic L, then the function and its variable in braces, then an integral from 0 to infinity of f(t) multiplied by e to the power minus s t, with a thin space before dt. The right-hand side is the integral that defines the transform; the left-hand side names the operator and the new variable s. If the L looks like a plain italic letter instead of a script capital, the command was typed outside math mode or as plain text (see troubleshooting).
Troubleshooting
The braces around f(t) do not print
A bare { is a LaTeX grouping character and does not print. Write \{ and \}.
Undefined control sequence for \mathscr
\mathscr is not defined in a base document. Load \usepackage{mathrsfs}, which per CTAN provides \mathscr and does not overwrite \mathcal. By contrast, CTAN notes that the separate calrsfs package does overwrite \mathcal, which would change every calligraphic letter in your document.
Math font style not found
Overleaf documents that some math font styles need \usepackage{amssymb} in the preamble. Add it if a font command is reported as undefined.
Missing $ inserted
Math commands only work in math mode. Put them between $ ... $, in \[ ... \], or in an equation environment.
Free alternatives (table)
All options below are written the same way; only the glyph and package change.
| Source | Needs | Note (source) |
|---|---|---|
\mathcal{L} |
Nothing extra for the basic form | Calligraphic L; documented on the Overleaf mathematical-fonts page. |
\mathscr{L} |
mathrsfs |
Raph Smith’s Formal Script font; leaves \mathcal intact (CTAN). |
\mathrm{L} |
Nothing extra | Upright roman L, for journals that prefer a non-script operator letter; check the journal’s author guidelines. |
FAQ
What is the LaTeX symbol for the Laplace transform?
The common convention is \mathcal{L}, for example \mathcal{L}\{f(t)\}. LaTeX has no built-in dedicated Laplace command.
How do I write the inverse Laplace transform?
Add a superscript: \mathcal{L}^{-1}\{F(s)\}.
How do I write the full Laplace transform formula?
Use \mathcal{L}\{f(t)\}(s) = \int_{0}^{\infty} f(t)\,e^{-st}\,dt in display math (step 3).
Where else do Greek letters and operators appear?
See our guides on Greek symbol commands in LaTeX, writing vectors in LaTeX and the product symbol in LaTeX. For the statistical distribution of the same name, see the Laplace distribution.
Sources (official docs)
- Overleaf documentation, Mathematical fonts, read 2026-11-08: overleaf.com/learn/latex/Mathematical_fonts
- Overleaf documentation, Integrals, sums and limits, read 2026-11-08: overleaf.com/learn/latex/Integrals,_sums_and_limits
- Overleaf documentation, Operators, read 2026-11-08: overleaf.com/learn/latex/Operators
- Overleaf documentation, List of Greek letters and math symbols, read 2026-11-08: overleaf.com/learn/latex/List_of_Greek_letters_and_math_symbols
- CTAN, mathrsfs package, read 2026-11-08: ctan.org/pkg/mathrsfs
