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numapprox

 laurent
 Laurent series expansion

 Calling Sequence laurent(f, x=a, n) laurent(f, x, n)

Parameters

 f - expression representing the function to be expanded x - the variable of expansion a - the point about which to expand n - (optional) non-negative integer stating the order of expansion

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{numapprox}\right):$
 > $\mathrm{laurent}\left(\frac{1}{x\mathrm{sin}\left(x\right)},x=0\right)$
 ${{x}}^{{-}{2}}{+}\frac{{1}}{{6}}{+}\frac{{7}}{{360}}{}{{x}}^{{2}}{+}{\mathrm{O}}\left({{x}}^{{4}}\right)$ (1)
 > $\mathrm{laurent}\left(\mathrm{Γ}\left(x\right),x,2\right)$
 ${{x}}^{{-}{1}}{-}{\mathrm{γ}}{+}\left(\frac{{1}}{{12}}{}{{\mathrm{π}}}^{{2}}{+}\frac{{1}}{{2}}{}{{\mathrm{γ}}}^{{2}}\right){}{x}{+}{\mathrm{O}}\left({{x}}^{{2}}\right)$ (2)
 > $r≔\frac{{x}^{2}+2x-3}{{x}^{4}-11{x}^{3}+42{x}^{2}-68x+40}$
 ${r}{≔}\frac{{{x}}^{{2}}{+}{2}{}{x}{-}{3}}{{{x}}^{{4}}{-}{11}{}{{x}}^{{3}}{+}{42}{}{{x}}^{{2}}{-}{68}{}{x}{+}{40}}$ (3)
 > $\mathrm{laurent}\left(r,x=2\right)$
 ${-}\frac{{5}}{{3}}{}{\left({x}{-}{2}\right)}^{{-}{3}}{-}\frac{{23}}{{9}}{}{\left({x}{-}{2}\right)}^{{-}{2}}{-}\frac{{32}}{{27}}{}{\left({x}{-}{2}\right)}^{{-}{1}}{-}\frac{{32}}{{81}}{-}\frac{{32}}{{243}}{}\left({x}{-}{2}\right){-}\frac{{32}}{{729}}{}{\left({x}{-}{2}\right)}^{{2}}{+}{\mathrm{O}}\left({\left({x}{-}{2}\right)}^{{3}}\right)$ (4)