RESIDUE: A residue package

REDUCE

16.59 RESIDUE: A residue package

This package supports the calculation of residues of arbitrary expressions.

Author: Wolfram Koepf.

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

∮ --1- Rze=sa f(z) = 2 πi f(z) dz ,

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

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

∞ ∑ k f(z) = ak (z − a) , k=− ∞

then

Res f(z ) = a −1 . z=a

(16.94)

If a = ∞, one deﬁnes on the other hand

Res f(z) = − a−1 z=∞

(16.95)

for given Laurent representation

∑∞ 1 f(z) = ak zk-. k=−∞

The package is loaded by the statement

1: load residue;

It contains two REDUCE operators:

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.
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 taylor package in connection with representations (16.94)–(16.95).

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

Note that the residues of factorial and Γ function terms are not yet supported.

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