−∞∑∞z3−n31=2πi⋅∑res f(z0) for z0 in the upper half-plane
1) z0=n
res1 f(z0)=z→nlim[f(z)(z−n)]=z→nlimz2+zn+n21=3n21
2) z0=n(−0.5+0.5i3)
res2 f(z0)=z→z0lim[f(z)(z−z0)]=z→z0limz2+zz0+z021=3z021=3n2(−0.5+0.5i3)21
3) z0=n(−0.5−0.5i3) is not in the upper half-plane
−∞∑∞z3−n31=3n22πi(1+(−0.5+0.5i3)21)
Comments