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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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