Residue

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Residue[f, {z, z0}]

gives the residue of f at the isolated singularity z = z0 -- the coefficient of (z - z0)^-1 in the Laurent expansion of f.

Notes

Computed by power-series expansion, so a residue is found only where f admits a Laurent series at z0. Returns unevaluated at branch points (fractional-power expansions) and when no series can be produced. See NResidue for a numerical alternative that also handles essential singularities.

Examples (19)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (9)

In[1]:= Residue[1/z, {z, 0}]
Out[1]= 1

In[2]:= Residue[1/z^2, {z, 0}]
Out[2]= 0

In[3]:= Residue[1/Sin[z]^5, {z, 0}]
Out[3]= 3/8

Order-2 pole

In[4]:= Residue[(z + 1)/(z^2 (z - 2)), {z, 0}]
Out[4]= -3/4

Complex pole

In[5]:= Residue[1/(z^2 + 1), {z, I}]
Out[5]= -1/2*I

Algebraic pole

In[6]:= Residue[x^3/(x^4 - 2), {x, 2^(1/4)}]
Out[6]= 1/4

Unknown numerator

In[7]:= Residue[f[z]/z^5, {z, 0}]
Out[7]= 1/24 Derivative[4][f][0]

In[8]:= Residue[Zeta[z]/(z - 1)^10, {z, 1}]
Out[8]= -1/362880 StieltjesGamma[9]

Branch point

In[9]:= Residue[1/Sqrt[z], {z, 0}]
Out[9]= Residue[1/Sqrt[z], {z, 0}]

Applications (10)

A simple pole at the origin

In[10]:= Residue[1/z, {z, 0}]
Out[10]= 1

Simple pole at a complex location, via P(z0)/Q'(z0)

In[11]:= Residue[1/(z^2 + 1), {z, I}]
Out[11]= -1/2*I

An algebraic pole location, kept in one radical spelling

In[12]:= Residue[1/(z^4 + 1), {z, Exp[I Pi/4]}]
Out[12]= -1/4 (-1)^(1/4)

Fifth-order pole of a transcendental integrand

In[13]:= Residue[1/Sin[z]^5, {z, 0}]
Out[13]= 3/8

The summation kernel Pi Cot[Pi z] at a pole of the rational factor

In[14]:= Residue[Pi Cot[Pi z]/(z^2 + 1), {z, I}]
Out[14]= -1/2 Pi Coth[Pi]

Residue 1 at the only pole of Zeta

In[15]:= Residue[Zeta[s], {s, 1}]
Out[15]= 1

Symbolic double pole, self-validated across expansion orders

In[16]:= Residue[Exp[I z]/(z^2 + a^2)^2, {z, I a}]
Out[16]= ((-1/4*I) E^(-a))/a^3 + ((-1/4*I) E^(-a))/a^2

Fractional power times a double pole, via the dropped-pole guard

In[17]:= Residue[z^(1/3)/(1 + z^2)^2, {z, I}]
Out[17]= (-1/6*I) I^(1/3)

A branch point has no residue, so the call stays unevaluated

In[18]:= Residue[1/Sqrt[z], {z, 0}]
Out[18]= Residue[1/Sqrt[z], {z, 0}]

Limitation: Series has no expansion of Gamma at its poles, so this stays unevaluated; NResidue gives -0.166667

In[19]:= Residue[Gamma[z], {z, -3}]
Out[19]= Residue[Gamma[z], {z, -3}]

Algorithm

residue.c -- Residue[expr, {z, z0}], the symbolic residue.

The residue of f at an isolated singularity z = z0 is the coefficient of (z - z0)^-1 in the Laurent expansion of f. We obtain it directly from the series engine: expand f to order (z - z0)^0 (which always spans the -1 term, however deep the pole), then read the coefficient at exponent -1 out of the resulting SeriesData[z, z0, {coefs}, nmin, nmax, den].

A residue is well defined only for an ordinary Laurent expansion (den == 1). A fractional-power (Puiseux) expansion, den > 1, signals a branch point, where the residue is undefined -- we leave the call unevaluated, matching Mathematica (e.g. Residue[1/Sqrt[z], {z, 0}]).

Algebraic pole locations.  The series engine decides whether z0 is a pole by

evaluating the denominator there and testing it against zero; but for a pole whose location is a SUM of radicals (e.g. z0 = -2 + Sqrt[3], a root of 1 + 4 z + z^2), Denominator(z0) is an expression like 1 + 4 (-2 + Sqrt[3]) + (-2 + Sqrt[3])^2 that does not auto-simplify to 0, so

the pole is missed and the residue wrongly comes out 0.  We defeat this by

expanding about z0 EXPLICITLY: substitute z -> z0 + w, then Expand the denominator of the result -- polynomial expansion collapses the radical arithmetic (Sqrt[3]^2 -> 3, ...) so the vanishing constant term becomes a

literal 0 and the w-factor of the pole is exposed.  Reading the (z-z0)^-1

coefficient is then a plain Series-at-0 of the expanded form.

Implementation notes

Algorithm. builtin_residue handles only the two-argument form Residue[f, {z, z0}] (fewer arguments emit Residue::argm; a non-symbol z or a malformed spec leaves the call unevaluated) and hands it to residue_compute, which reads the coefficient of (z - z0)^-1 out of a Laurent expansion. The integrand is first classified by residue_is_rational_in (is Together[f] a ratio of polynomials in z?). For a rational f, residue_simple_pole tries the fast path P(z0)/Q'(z0) (P/Q = Together[f], Q(z0) == 0 and Q'(z0) != 0 decided by PossibleZeroQ), which never builds a series. Otherwise residue_shift_form substitutes z -> z0 + w (a private symbol Residue`$w) and Expands the shifted numerator and denominator separately, so Q's constant term collapses to a literal 0 and an algebraic z0 keeps one radical spelling. Transcendental integrands (Cot, Zeta, unknown f[z]/z^n) are expanded directly about z0, so the series engine's own Laurent knowledge applies.

In both cases the coefficient comes from residue_extract, which is self-validating since v0.268. An order-0 probe (residue_coeff_once) learns the pole order m and short-circuits the analytic (residue 0) and branch-point (den > 1, unevaluated) cases. residue_coeff_resolved then expands from order m - 1, raising the order by 3 while the O-term still hides exponent -1. Successive orders are compared (residue_coeffs_agree, a PossibleZeroQ of the difference), and a coefficient is accepted only when two consecutive orders agree. A padded Series can return a -1 coefficient that is present but inaccurate at a higher-order pole: the cofactor is truncated, so product-rule cross terms drop out, and this happened at symbolic double poles such as E^(I z)/(z^2 + a^2)^2 at I a. The agreement test catches that case. A second guard covers a dropped pole. When the direct expansion reports 0 but residue_has_pole_at finds that Denominator[Together[f]] vanishes at z0, the function re-expands f /. z -> z0 + w about w = 0. There z^p becomes the analytic binomial (z0 + w)^p, which fixes the silent 0 previously returned for z^(1/3)/(1 + z^2)^2 at z = I.

Data structures. Expr trees throughout. The helpers residue_eval1/residue_eval2 build and evaluate calls to Together, Numerator, Denominator, Expand, D, ReplaceAll and Series. The expansion is a SeriesData[z, z0, {coefs}, nmin, nmax, den], and the residue is the coefficient at index -1 - nmin.

Complexity / limits. The simple-pole path costs one differentiation and two substitutions. The series path costs a few Laurent expansions. The resolve loop gives up past order 256, and the agreement loop is bounded at order 64 / 12 iterations, returning the last estimate. The residue is defined only for an ordinary Laurent expansion, so a Puiseux expansion (a branch point, as in 1/Sqrt[z] at 0) is left unevaluated. So is any point where Series itself has no expansion. The essential singularity E^(1/z) at 0 and the poles of Gamma at the non-positive integers fall in this class (Series[Gamma[z], {z, 0, n}] is itself unevaluated). For those cases, use the numerical NResidue.

Attributes: Protected.

References

See also: NResidue, Together, Zeta, Series

• Source: src/calculus/residue.c
• Specification: docs/spec/builtins/calculus.md
• Tests: tests/test_residue.c

Notes & additional examples

Notes

• Residue[f, {z, z0}] is the coefficient of (z - z0)^-1 in the Laurent expansion of f at z0. An analytic point gives 0. Poles of any order, at numeric or symbolic centres, are handled.
• For higher-order poles the coefficient is accepted only after two consecutive expansion orders agree. This closes the silent wrong answers at symbolic double and triple poles that existed before v0.268. Those wrong residues also fed Integrate[..., Method -> "Residue"].
• A branch point (a fractional-power expansion) is left unevaluated, matching Mathematica. An essential singularity (E^(1/z) at 0) and the poles of Gamma are also left unevaluated, because Series has no expansion there. Mathematica returns 1 and (-1)^n/n! for these. Use NResidue for a numerical value.
• Only the two-argument form is supported. Attributes: Protected.
• See also: NResidue, Series, SeriesCoefficient, Integrate (contour / residue-theorem definite integration).