How can I define a mathematical function as a LaTeX macro?

Question

I am attempting to create a usable function within LaTeX such that its inputs are transformed mathematically. However, I cannot find any documentation that can perform such mathematical function definition.

Here is an example of what I am seeking:

Having defined a function entitled \myMathFunction which accepts only one parameter, the follow usage:

\myMathFunction{2}

should output 6 if the function is defined as follows: ({#1} * 5) - {#1}^2

Where {#1} is the parameter, and in the case of my example, is equal to 2.

Additionally, the function should be usable within a PSTricks plot. E.g.

\psplot[
    algebraic,
    linecolor   = red,
    linewidth   = 1pt
]{0}{TwoPi}
{\myMathFunction{x}}

Answer 1

A calculated table and the plotted curve:

\documentclass{article}
\usepackage{pst-plot}
\usepackage{xparse}
\ExplSyntaxOn%% allow _ : as a letter
\newcommand\myMathFunction[1]{% 
  \ifx#1x ((#1) * 5) - (#1)^2  
  \else \fp_to_decimal:n {((#1) * 5) - (#1)^2} \fi}
\ExplSyntaxOff
\begin{document}
\begin{tabular}{ r r }
$x$ & $f(x)$ \\\hline
0 & \myMathFunction{0}\\
1 & \myMathFunction{1}\\
2 & \myMathFunction{2}\\
3.3 & \myMathFunction{3.3}\\
4 & \myMathFunction{4}\\
5 & \myMathFunction{5}\\\hline
\end{tabular}
%
\pspicture[shift=*](-1,0)(6,7)
\psaxes{->}(5.5,6)
\psplot[algebraic,linecolor=red,linewidth=1.5pt]{0}{5}{\myMathFunction{x}}
\endpspicture
\end{document}

Answer 2

Remarks

I used the powerful LaTeX3 featureset l3fp, which is automatically loaded by xparse.

Implementation

\documentclass{article}
\usepackage{xparse}
\ExplSyntaxOn
\NewDocumentCommand{\myMathFunction}{m}
    { \fp_to_decimal:n {((#1) * 5) - (#1)^2} }
\ExplSyntaxOff
\begin{document}
\myMathFunction{2}
\end{document}

Answer 3

Here is a TikZ/PGF solution.

I'm not sure how it compares to the l3fp approach, but it definitely offers more flexibility than a low-level TeX approach because

• it works in fixed-point arithmetic, not just with integers, and
• by using the right PGFkeys, you can easily customise how the result should be printed (trailing zeros, scientific notation, etc.). I refer you to section 66: Number printing of the PFG manual for more details on that.

\documentclass{article}

\usepackage{tikz}

\begin{document}

% definiton
\newcommand\myMathFunction[1]%
{%
    \pgfmathparse{5*#1-(#1)^2}%
    \pgfmathprintnumber[fixed,precision=3]{\pgfmathresult}%
}

% macro calls
\myMathFunction{2} \quad
\myMathFunction{45} \quad
\myMathFunction{-56}

\end{document}

Answer 4

In good old (Plain) TeX, i.e., without LaTeX, with the proper TeX syntax:

\documentclass{article}

\begin{document}

\newcount\pom % temporary
\newcount\kw % square
\newcount\first % first

\def\myMathFunction#1{\pom#1  \first\pom \kw\pom  
\multiply\kw by\pom \multiply\first by5 
\advance\first by-\kw \the\first}

\myMathFunction{2}


And an example of loop:

\newcount\n
\n-10
\loop \ifnum\n<10 $f(\the\n)=\myMathFunction{\n}$ \advance\n by1 \repeat

\end{document}

We could do it using just two counters, but this solution is easier to understand. Numbers are integers, with the absolute value less than 2^31 on all stages of computing.

Answer 5

Here's a solution using LuaLaTeX. The MWE provides a LaTeX-side macro called \MyMathFunction that interfaces with a Lua-side function called mymathfunction; the latter does the actual computations.

% !TEX TS-program = lualatex
\documentclass{article}
\usepackage{amsmath}    % for '\ensuremath' macro
\usepackage{luacode}    % for 'luacode' environment

% Lua-side code
\begin{luacode}
   function mymathfunction( x )
      return 5*x - x^2  -- specify the function you need
   end
\end{luacode}

%TeX-side macro that invokes the Lua function
\newcommand{\MyMathFunction}[1]{%
    \ensuremath{\directlua{tex.sprint( mymathfunction ( #1 ) )}}}

\begin{document}
The value of my math function evaluated at 2 is \MyMathFunction{2}.

\newcommand\myval{6}  % macro that contains parameter value
The value of my math function evaluated at \myval\ is \MyMathFunction{\myval}.

\end{document}

Observe that the value of \myval could itself be the result of a calculation done in Lua. For example, you could execute

\newcommand\myval{ \directlua{ tex.sprint( math.exp(1) ) } }

to set \myval equal to the base of the natural logarithms. And, one could pass the result of a calculation done in Lua directly to \MyMathFunction, i.e., without setting up a macro called \myval to contain the result of the calculation. For instance,

\MyMathFunction{ \directlua{ tex.sprint( math.sqrt(2) ) }}

will evaluate \MyMathFunction at the square root of 2.

Answer 6

Using PythonTeX:

\documentclass{article}
\usepackage{pythontex}

\newcommand{\MyMathFunction}[1]{\py{2*(#1)**3 - 7}}

\begin{document}
\MyMathFunction{2}
\end{document}

Suppose the above is the content of the file my_math_fun_with_python.tex.

You run LeTeX on the file:

latex my_math_fun_with_latex

then you run pythontex:

pythontex.py my_math_fun_with_latex

That will execute the python code in the file, and save the results.

Than you run LaTeX again:

latex my_math_fun_with_latex

Answer 7

The fp package provides a LaTeX2e-like interface to arithmetic:

\documentclass{article}
\usepackage[nomessages]{fp}% http://ctan.org/pkg/fp
\newcommand{\myMathFunction}[1]
    {\FPeval\result{trunc(((#1) * 5) - (#1)^2:0)}\result}
\begin{document}
\verb|\myMathFunction{2}|: \myMathFunction{2}
\end{document}

Answer 8

A much more general solution with the omnipotent Mathematica, LaTeX and PSTricks. In this answer I show how to

• evaluate a function,
• factor it into the product of linear real binomials,
• plot its graph.

Step 1

Purchase Mathematica Home Edition (about USD $250), download and install it.

Step 2

Create a template named main.mtex (for example). The extension .mtex is important because Mathematica uses "convention over configuration" philosophy.

% main.mtex
\documentclass[preview,border=12pt,12pt]{standalone}
\usepackage{amsmath}
\usepackage{pst-plot}
\usepackage{verbatim}
\begin{comment}
<* 
    f[x_] := x^2 - 5 x + 6;
    rep:= x_Times -> Infix[x, "*"];
*>
\end{comment}


\begin{document}
\section*{Problem}
Let $ y = f(x) = <* f[x] *>$,
\begin{enumerate}
    \item factor $f(x)$
    \item evaluate $f(\tfrac 1 2)$
    \item plot its graph.
\end{enumerate}

\section*{Solution}
\begin{enumerate}
    \item $f(x) = <* Factor[f[x]] *>$
    \item $f(\tfrac 1 2)= <* f[1/2] *> $
    \item \hspace{20pt}
    \begin{pspicture}(0,-1)(5,3)
        \psaxes{->}(0,0)(0,-1)(4.5,2.5)[$x$,0][$y$,90]
        \psplot[algebraic,linecolor=blue]{1}{4}{<* f[x] /. rep *>}
    \end{pspicture}
\end{enumerate}

\end{document}

Everything in <* ... *> contains Mathematica commands and they will be replaced by invoking Splice function that will be explained shortly.

Step 3

Create an new Mathematica document named Splicer.nb (for example) as follows.

Splice["C:\\Users\\Donut E. Knot\\Documents\\main.mtex"];

And execute the command by pressing Shift+Enter. Mathematica will generate a file main.tex containing code as follows.

% main.tex
\documentclass[preview,border=12pt,12pt]{standalone}
\usepackage{amsmath}
\usepackage{pst-plot}
\usepackage{verbatim}
\begin{comment}
\text{Null}
\end{comment}


\begin{document}
\section*{Problem}
Let $ y = f(x) = x^2-5 x+6$,
\begin{enumerate}
    \item factor $f(x)$
    \item evaluate $f(\tfrac 1 2)$
    \item plot its graph.
\end{enumerate}

\section*{Solution}
\begin{enumerate}
    \item $f(x) = (x-3) (x-2)$
    \item $f(\tfrac 1 2)= \frac{15}{4} $
    \item \hspace{20pt}
    \begin{pspicture}(0,-1)(5,3)
        \psaxes{->}(0,0)(0,-1)(4.5,2.5)[$x$,0][$y$,90]
        \psplot[algebraic,linecolor=blue]{1}{4}{x^2+-5*x+6}
    \end{pspicture}
\end{enumerate}

\end{document}

Step 4

Compile the main.tex with latex-dvips-ps2pdf as it contains PSTricks code. The output is...

Step 5

Done. :-)

Answer 9

My package 'calculus' (installed together with 'calculator') defines some commands to declare mathematical functions. Unfortunately these functions can not be drawn using PSTricks, but you can draw it using my package 'xpicture".

Your example can be defined typing

\newqpoly{\myMathFunction}{0}{5}{-1}

(this declare the quadratic polynomial p(x)=0+5x-x^2). Then, you can compute this function and its derivative at x=2 by typing

\myMathFunction{2}{\sol}{\Dsol}

(\sol stores p(2); \Dsol stores p'(2)). Try

If $p(x)=-x^2+5x$, then $p(2)=\sol$ and $p'(2)=\Dsol$.

This code draw the graph:

\setlength{\unitlength}{0.75cm}
\begin{Picture}(-0.5,-10.5)(6.5,10.5)
   \cartesianaxes(0,-10)(\numberTWOPI,7)
   \PlotFunction{\myMathFunction}{0}{\numberTWOPI}
\end{Picture}

Answer 10

Another solution with Maxima, a free computer algebra system. Note: Only for Windows users.

Step 1

Download and install Maxima.

Step 2

Create a batch file named cas.bat (for example) as follows.

rem cas.bat
echo off
set PATH=%PATH%;"C:\Program Files (x86)\Maxima-5.31.2\bin"
maxima --very-quiet -r %1 > solution.tex

Save the batch in the same directory in which your input file below exists. It is just for the sake of simplicity.

Step 3

Create the input file named main.tex (for example) as follows.

% main.tex
\documentclass[preview,border=12pt,12pt]{standalone}
\usepackage{amsmath}

\def\f(#1){(#1)^2-5*(#1)+6}


\begin{document} 

\section{Problem}
Evaluate $\f(x)$ for $x=\frac 1 2$.

\section{Solution}
\immediate\write18{cas "x: 1/2\string$ tex(\f(x))\string$"}

\input{solution}

\end{document}

Step 4

Compile the input file with pdflatex -shell-escape main and you will get a nice output as follows.

Miscellaneous

Apparently the output of Maxima is as follows.

solution.tex
$${{15}\over{4}}$$

Answer 11

For an expression with the basic operations and the square root extraction [and also logic operators and conditional evaluation] you can use \xintNewExpr or \xintNewFloatExpr from the xintexpr package to create a (completely expandable) mathematical function.

The package core routines compute exactly, with an arbitrary number of digits, and natively with fractions too. The floating point routines compute with a user-specified precision. Default is with 16 digits.

The main limitation at the time of writing is that currently sqrt is the only non-rational operation permitted. Powers must be with integral exponents.

Macros defined via \xintNewExpr or \xintNewFloatExpr may be nested one within the other. This feature was broken from release 1.09i to release 1.09ka, version 1.09kb [just released at the time of writing] reverted the bug which had been introduced in 1.09i.

\documentclass{article}
\usepackage[margin=1cm]{geometry}
\usepackage{xintexpr}
\xintNewExpr \MathFunction [1] {#1 * 5 - #1^2}

\xintNewExpr \MathFunctionRounded [1] {round(#1*5 - #1^2,12)}

\xintNewExpr \MathFunctionRoundedBis [2] {round(#1*5 - #1^2,#2)}

\xintNewFloatExpr \MathFunctionFloat [1] {#1 * 5 - #1^2}

\begin{document}\thispagestyle{empty}
\begin{verbatim}
\xintNewExpr \MathFunction [1] {#1 * 5 - #1^2}
\xintNewExpr \MathFunctionRounded [1] {round(#1*5 - #1^2,12)}
\xintNewExpr \MathFunctionRoundedBis [2] {round(#1*5 - #1^2,#2)}
\xintNewFloatExpr \MathFunctionFloat [1] {#1 * 5 - #1^2}
\end{verbatim}

\ttfamily

output in xint own fraction format:\newline
\verb|\MathFunction {12344321.56789876}->|\MathFunction {12344321.56789876}

\medskip

output as a fixed point number (12 digits after decimal mark):\newline
\verb|\MathFunctionRounded {12344321.56789876}->|\MathFunctionRounded
{12344321.56789876}

\medskip

output as a fixed point number (20 digits after decimal mark):\newline
\verb|\MathFunctionRoundedBis {12344321.56789876}{20}->|\MathFunctionRoundedBis
{12344321.56789876}{20}

\medskip

floating point number (16 digits of precision by default)\newline
\verb|\MathFunctionFloat {12344321.56789876}->|\MathFunctionFloat {12344321.56789876}

\medskip

floating point number (20 digits of precision; macro need not be redefined)\newline
\verb|\xintDigits := 20;|\xintDigits := 20;\newline
\verb|\MathFunctionFloat {12344321.56789876}->|\MathFunctionFloat {12344321.56789876}


\medskip

nesting of macros:\newline
\verb|\MathFunctionFloat {\MathFunctionFloat {12344321.56789876}}->|\MathFunctionFloat {\MathFunctionFloat {12344321.56789876}}

\medskip

\noindent\verb|\MathFunction {\MathFunction {12344321.56789876}}->|\newline
\MathFunction {\MathFunction {12344321.56789876}}

\medskip

\verb|\MathFunction has meaning| \newline
\meaning\MathFunction

\end{document}