7.15 RESIDUE and POLEORDER Operators

The residue \(\mathop {\mathrm {Res}}\limits _{z=a} f(z)\) of a function \(f(z)\) at the point \(a\in \mathbb {C}\) is defined as \[ \mathop {\mathrm {Res}}\limits _{z=a} f(z)= \frac {1}{2 \pi i}\oint f(z)\,dz \;, \] with integration along a closed curve around \(z=a\) with winding number 1.

If \(f(z)\) is given by a Laurent series expansion at \(z=a\) \[ f(z)=\sum _{k=-\infty }^\infty a_k\,(z-a)^k \;, \] then \begin {equation} \mathop {\mathrm {Res}}\limits _{z=a} f(z)=a_{-1} \;. \label {eq:Laurent} \end {equation} If \(a=\infty \), one defines on the other hand \begin {equation} \mathop {\mathrm {Res}}\limits _{z=\infty } f(z)=-a_{-1} \label {eq:Laurent2} \end {equation} for given Laurent representation \[ f(z)=\sum _{k=-\infty }^\infty a_k\,\frac {1}{z^k} \;. \] 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 (\ref {eq:Laurent})–(\ref {eq:Laurent2}).

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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