Computes the algebraic residue of the expression f for the variable x around the point a. The residue is defined as the coefficient of $\frac{1}{x-a}$ in the Laurent series expansion of f.

Maple compute the residue by successively performing series expansions at $x=a$, by default of orders $2,3,6,11,18,27$, and then extracts the coefficient of $\frac{1}{x-a}$. If this is unsuccessful, for example, because $f$ does not have a Laurent expansion, or because order $27$ is still too small, the residue command returns unevaluated.

The maximal order of series expansions being tried can be raised by providing a positive integer n as third argument.

$\mathrm{residue}\left(\mathrm{\zeta }\left(s\right)\,s=1\right)$

$1$

$\mathrm{residue}\left(\frac{\mathrm{\Psi }\left(x\right)\mathrm{\Gamma }\left(x\right)}{x}\,x=0\right)$

$\frac{{\mathrm{\pi }}^2}{12}+\frac{{\mathrm{\gamma }}^2}{2}$

$\mathrm{residue}\left(\exp \left(x\right)\,x=1\right)$

$0$

$\mathrm{residue}\left(\arcsin \left(x\right)\,x=1\right)$

$\mathrm{residue}\left(\arcsin \left(x\right)\,x=1\right)$

$\mathrm{series}\left(\arcsin \left(x\right)\,x=1\,3\right)$

$\frac{\mathrm{\pi }}{2}-\mathrm{I}\sqrt{2}\mathrm{csgn}\left(\mathrm{I}\left(x-1\right)\right)\sqrt{x-1}+\frac{\mathrm{I}\sqrt{2}\mathrm{csgn}\left(\mathrm{I}\left(x-1\right)\right){\left(x-1\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{12}-\frac{3\mathrm{I}\sqrt{2}\mathrm{csgn}\left(\mathrm{I}\left(x-1\right)\right){\left(x-1\right)}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{160}+\mathrm{O}\left({\left(x-1\right)}^{\raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$

$\mathrm{residue}\left(\frac{1}{z^{27}{\left(1-z\right)}^3}\,z=0\right)$

$378$

$\mathrm{residue}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\right)$

$\mathrm{residue}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\right)$

$\mathrm{residue}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\,27\right)$

$\mathrm{residue}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\,27\right)$

$\mathrm{series}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\,27\right)$

$z^{\mathrm{-28}}+3z^{\mathrm{-27}}+6z^{\mathrm{-26}}+10z^{\mathrm{-25}}+15z^{\mathrm{-24}}+21z^{\mathrm{-23}}+28z^{\mathrm{-22}}+36z^{\mathrm{-21}}+45z^{\mathrm{-20}}+55z^{\mathrm{-19}}+66z^{\mathrm{-18}}+78z^{\mathrm{-17}}+91z^{\mathrm{-16}}+105z^{\mathrm{-15}}+120z^{\mathrm{-14}}+136z^{\mathrm{-13}}+153z^{\mathrm{-12}}+171z^{\mathrm{-11}}+190z^{\mathrm{-10}}+210z^{\mathrm{-9}}+231z^{\mathrm{-8}}+253z^{\mathrm{-7}}+276z^{\mathrm{-6}}+300z^{\mathrm{-5}}+325z^{\mathrm{-4}}+351z^{\mathrm{-3}}+378z^{\mathrm{-2}}+O\left(z^{\mathrm{-1}}\right)$

$\mathrm{residue}\left(\frac{1}{z^{28}{\left(1-z\right)}^3}\,z=0\,28\right)$

$406$

The residue command was updated in Maple 2019.

The n parameter was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.