REDUCE
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7.15 RESIDUE and POLEORDER Operators

The residue Resz=af(z) of a function f(z) at the point a ∈ℂ is defined as

∮ Res f(z) = -1-- f(z)dz , z=a 2πi

with integration along a closed curve around z = a with winding number 1.

If f(z) is given by a Laurent series expansion at z = a

∞ ∑ k f(z) = ak(z − a ) , k=− ∞

then

Rze=sa f (z) = a−1 .
(7.1)

If a = ∞, one defines on the other hand

Res f (z) = − a z=∞ −1
(7.2)

for given Laurent representation

∞∑ 1 f(z) = ak -k-. k=−∞ z

The operator residue(f,z,a) determines the residue of f at the point z = a if f is meromorphic at z = a. The calculation of residues at essential singularities of f is not supported, as are the residues of factorial terms.2

poleorder(f,z,a) determines the pole order of f at the point z = a if f is meromorphic at z = a.

Note that both functions use the operator taylor in connection with representations (7.1)–(7.2).

Here are some examples: 

2: residue(x/(x^2-2),x,sqrt(2));

 1
---
 2

3: poleorder(x/(x^2-2),x,sqrt(2));

1

4: residue(sin(x)/(x^2-2),x,sqrt(2));

 sqrt(2)*sin(sqrt(2))
----------------------
          4

5: poleorder(sin(x)/(x^2-2),x,sqrt(2));

1

6: residue(1/(x-1)^m/(x-2)^2,x,2);

 - m

7: poleorder(1/(x-1)/(x-2)^2,x,2);

2

8: residue(sin(x)/x^2,x,0);

1

9: poleorder(sin(x)/x^2,x,0);

1

10: residue((1+x^2)/(1-x^2),x,1);

-1

11: poleorder((1+x^2)/(1-x^2),x,1);

1
                                                                     

                                                                     

12: residue((1+x^2)/(1-x^2),x,-1);

1

13: poleorder((1+x^2)/(1-x^2),x,-1);

1

14: residue(tan(x),x,pi/2);

-1

15: poleorder(tan(x),x,pi/2);

1

16: residue((x^n-y^n)/(x-y),x,y);

0

17: poleorder((x^n-y^n)/(x-y),x,y);

0

18: residue((x^n-y^n)/(x-y)^2,x,y);

  n
 y *n
------
  y

19: poleorder((x^n-y^n)/(x-y)^2,x,y);

1

20: residue(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);

-2

21: poleorder(tan(x)/sec(x-pi/2)+1/cos(x),x,pi/2);

1

22: for k:=1:2 sum residue((a+b*x+c*x^2)/(d+e*x+f*x^2),x,
    part(part(solve(d+e*x+f*x^2,x),k),2));

 b*f - c*e
-----------
                                                                     

                                                                     
     2
    f

23: residue(x^3/sin(1/x)^2,x,infinity);

  - 1
------
  15

24: residue(x^3*sin(1/x)^2,x,infinity);

-1

25: residue(gamma(x),x,-1);

-1

26: residue(psi(x),x,-1);

-1

27: on fullroots;

28: for k:=1:3 sum
28: residue((a+b*x+c*x^2+d*x^3)/(e+f*x+g*x^2+h*x^3),x,
28: part(part(solve(e+f*x+g*x^2+h*x^3,x),k),2));

0

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