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### 16.55 RESIDUE: A residue package

This package supports the calculation of residues of arbitrary expressions.

Author: Wolfram Koepf.

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

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

then

 (16.94)

If a = , one defines on the other hand

 (16.95)

for given Laurent representation

The package is loaded by the statement

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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