- #1
nhrock3
- 415
- 0
[tex]\int_{|z|=3}^{nothing}\frac{dz}{z^3(z^{10}-2)}\\[/tex]
[tex]f=\frac{1}{z^3(z^{10}-2)}\\[/tex]
[tex]f(\frac{1}{z})=\frac{1}{(\frac{1}{z})^3((\frac{1}{z})^{10}-2)}\frac{z^{13}}{1-2z^{10}}=\\[/tex]
[tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=\frac{1}{z^2}\sum_{n=0}^{\infty}(2z^{10})^n\\[/tex]
[tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=-[res(f,inside|z|=3)+res(f,outside|z|=3)][/tex]
from the sum i get that there is no [tex]z^{-1}[/tex] member in the series
so the coefficient of [tex]z^{-1}[/tex] is zero
so the residiu of infinity is zero
but still all of my singular points are |z|=3
so the integral equals zero
??
did i solved it correctly
did i written every formula regarding the laws of residue correctly here
?
[tex]f=\frac{1}{z^3(z^{10}-2)}\\[/tex]
[tex]f(\frac{1}{z})=\frac{1}{(\frac{1}{z})^3((\frac{1}{z})^{10}-2)}\frac{z^{13}}{1-2z^{10}}=\\[/tex]
[tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=\frac{1}{z^2}\sum_{n=0}^{\infty}(2z^{10})^n\\[/tex]
[tex]res(f,\infty)= res(\frac{1}{z^2}f(\frac{1}{z}),0)=-[res(f,inside|z|=3)+res(f,outside|z|=3)][/tex]
from the sum i get that there is no [tex]z^{-1}[/tex] member in the series
so the coefficient of [tex]z^{-1}[/tex] is zero
so the residiu of infinity is zero
but still all of my singular points are |z|=3
so the integral equals zero
??
did i solved it correctly
did i written every formula regarding the laws of residue correctly here
?