The function laurent computes the Laurent series expansion of f, with respect to the variable x, about the point a, up to order n.

The laurent function is a restriction of the more general series function. See series for a complete explanation of the parameters.

If the result of the series function applied to the specified arguments is a Laurent series with finite principal part (i.e. only a finite number of non-negative powers appear in the series) then this result is returned; otherwise, an error-return occurs.

The command with(numapprox,laurent) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{laurent}\left(\frac{1}{x\sin \left(x\right)}\,x=0\right)$

$x^{\mathrm{-2}}+\frac{1}{6}+\frac{7}{360}{x}^2+O\left(x^4\right)$

$\mathrm{laurent}\left(\mathrm{\Gamma }\left(x\right)\,x\,2\right)$

$x^{\mathrm{-1}}-\mathrm{\gamma }+\left(\frac{{\mathrm{\pi }}^2}{12}+\frac{{\mathrm{\gamma }}^2}{2}\right)x+O\left(x^2\right)$

$r≔\frac{x^2+2x-3}{x^4-11x^3+42x^2-68x+40}$

$r≔\frac{x^2+2x-3}{x^4-11x^3+42x^2-68x+40}$

$\mathrm{laurent}\left(r\,x=2\right)$

$-\frac{5}{3}{\left(x-2\right)}^{\mathrm{-3}}-\frac{23}{9}{\left(x-2\right)}^{\mathrm{-2}}-\frac{32}{27}{\left(x-2\right)}^{\mathrm{-1}}-\frac{32}{81}-\frac{32}{243}\left(x-2\right)-\frac{32}{729}{\left(x-2\right)}^2+O\left({\left(x-2\right)}^3\right)$