residue - Maple Programming Help

residue

compute the algebraic residue of an expression

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

Parameters

 f - arbitrary algebraic expression x - variable a - algebraic value at which the residue is evaluated n - (optional) positive integer

Description

 • 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.

Examples

 > $\mathrm{residue}\left(\mathrm{\zeta }\left(s\right),s=1\right)$
 ${1}$ (1)
 > $\mathrm{residue}\left(\frac{\mathrm{\Psi }\left(x\right)\mathrm{\Gamma }\left(x\right)}{x},x=0\right)$
 $\frac{{{\mathrm{\pi }}}^{{2}}}{{12}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}}{{2}}$ (2)
 > $\mathrm{residue}\left(\mathrm{exp}\left(x\right),x=1\right)$
 ${0}$ (3)

In the following example, there is no Laurent expansion.

 > $\mathrm{residue}\left(\mathrm{arcsin}\left(x\right),x=1\right)$
 ${\mathrm{residue}}{}\left({\mathrm{arcsin}}{}\left({x}\right){,}{x}{=}{1}\right)$ (4)
 > $\mathrm{series}\left(\mathrm{arcsin}\left(x\right),x=1,3\right)$
 $\frac{{\mathrm{\pi }}}{{2}}{-}{I}{}\sqrt{{2}}{}{\mathrm{csgn}}{}\left({I}{}\left({x}{-}{1}\right)\right){}\sqrt{{x}{-}{1}}{+}\frac{{I}{}\sqrt{{2}}{}{\mathrm{csgn}}{}\left({I}{}\left({x}{-}{1}\right)\right){}{\left({x}{-}{1}\right)}^{{3}}{{2}}}}{{12}}{-}\frac{{3}{}{I}{}\sqrt{{2}}{}{\mathrm{csgn}}{}\left({I}{}\left({x}{-}{1}\right)\right){}{\left({x}{-}{1}\right)}^{{5}}{{2}}}}{{160}}{+}{\mathrm{O}}{}\left({\left({x}{-}{1}\right)}^{{7}}{{2}}}\right)$ (5)

In the next example, the default order is too small.

 > $\mathrm{residue}\left(\frac{1}{{z}^{27}{\left(1-z\right)}^{3}},z=0\right)$
 ${378}$ (6)
 > $\mathrm{residue}\left(\frac{1}{{z}^{28}{\left(1-z\right)}^{3}},z=0\right)$
 ${\mathrm{residue}}{}\left(\frac{{1}}{{{z}}^{{28}}{}{\left({1}{-}{z}\right)}^{{3}}}{,}{z}{=}{0}\right)$ (7)
 > $\mathrm{residue}\left(\frac{1}{{z}^{28}{\left(1-z\right)}^{3}},z=0,27\right)$
 ${\mathrm{residue}}{}\left(\frac{{1}}{{{z}}^{{28}}{}{\left({1}{-}{z}\right)}^{{3}}}{,}{z}{=}{0}{,}{27}\right)$ (8)

The reason is that the series expansion of order $27$ does not have enough terms:

 > $\mathrm{series}\left(\frac{1}{{z}^{28}{\left(1-z\right)}^{3}},z=0,27\right)$
 ${{z}}^{{-28}}{+}{3}{}{{z}}^{{-27}}{+}{6}{}{{z}}^{{-26}}{+}{10}{}{{z}}^{{-25}}{+}{15}{}{{z}}^{{-24}}{+}{21}{}{{z}}^{{-23}}{+}{28}{}{{z}}^{{-22}}{+}{36}{}{{z}}^{{-21}}{+}{45}{}{{z}}^{{-20}}{+}{55}{}{{z}}^{{-19}}{+}{66}{}{{z}}^{{-18}}{+}{78}{}{{z}}^{{-17}}{+}{91}{}{{z}}^{{-16}}{+}{105}{}{{z}}^{{-15}}{+}{120}{}{{z}}^{{-14}}{+}{136}{}{{z}}^{{-13}}{+}{153}{}{{z}}^{{-12}}{+}{171}{}{{z}}^{{-11}}{+}{190}{}{{z}}^{{-10}}{+}{210}{}{{z}}^{{-9}}{+}{231}{}{{z}}^{{-8}}{+}{253}{}{{z}}^{{-7}}{+}{276}{}{{z}}^{{-6}}{+}{300}{}{{z}}^{{-5}}{+}{325}{}{{z}}^{{-4}}{+}{351}{}{{z}}^{{-3}}{+}{378}{}{{z}}^{{-2}}{+}{O}{}\left({{z}}^{{-1}}\right)$ (9)

Raising the order to 28 helps:

 > $\mathrm{residue}\left(\frac{1}{{z}^{28}{\left(1-z\right)}^{3}},z=0,28\right)$
 ${406}$ (10)

Compatibility

 • The residue command was updated in Maple 2019.
 • The n parameter was introduced in Maple 2019.