# another complex question

• May 27th 2010, 11:30 PM
fredrick08
another complex question
$\displaystyle \int_{-\infty}^{\infty}\frac{1}{x^{6}+1}dx$

$\displaystyle =\int_{C}\frac{1}{z^{6}+1}dz$ i think.... a bit confused because there will be singularities in UHP and LHP...

$\displaystyle =\int_{C}\frac{1}{(z^{2}+1)(z^{4}-z^{2}+1)}dz$

$\displaystyle =\int_{C}\frac{1}{(z-i)(z+i)(z^{4}-z^{2}+1)}dz$

let z^4=z^2

$\displaystyle z^{2}=\frac{1\pm\sqrt{1-4}}{2}=\frac{1\pm\sqrt{3}\\i}{2}$

$\displaystyle \\\\z=\{-i,i,\frac{1}{2}(\sqrt{3}+i),\frac{1}{2}(\sqrt{3}-i),\frac{1}{2}(-\sqrt{3}+i),\frac{1}{2}(-\sqrt{3}-i)\}$

now from here... how the hell do i do the residues.. the singularities are all over the place???

• May 27th 2010, 11:36 PM
Drexel28
Quote:

Originally Posted by fredrick08
$\displaystyle \int_{-\infty}^{\infty}\frac{1}{x^{6}+1}dx$

$\displaystyle =\int_{C}\frac{1}{z^{6}+1}dz$ i think.... a bit confused because there will be singularities in UHP and LHP...

$\displaystyle =\int_{C}\frac{1}{(z^{2}+1)(z^{4}-z^{2}+1)}dz$

$\displaystyle =\int_{C}\frac{1}{(z-i)(z+i)(z^{4}-z^{2}+1)}dz$

let z^4=z^2

$\displaystyle z^{2}=\frac{1\pm\sqrt{1-4}}{2}=\frac{1\pm\sqrt{3}\\i}{2}$

$\displaystyle \\\\z=\{-i,i,\frac{1}{2}(\sqrt{3}+i),\frac{1}{2}(\sqrt{3}-i),\frac{1}{2}(-\sqrt{3}+i),\frac{1}{2}(-\sqrt{3}-i)\}$

now from here... how the hell do i do the residues.. the singularities are all over the place???

What do you mean how do you find the residues?? How else does one usually find them?
• May 27th 2010, 11:54 PM
chisigma
The poles of the function $\displaystyle f(z)= \frac{1}{1+z^{6}}$ are at $\displaystyle z= e^{\pm i \pi \frac{1+2k}{6}}$ , $\displaystyle k=0,1,2$. Only the residues of the poles with positive imaginary part give contribution of the integral that is...

$\displaystyle \int_{-\infty}^{+\infty} \frac{dx}{1+x^{6}} = 2 \pi i \sum_{k=0}^{2} r_{k}$ (1)

... where...

$\displaystyle r_{0} = \lim_{z \rightarrow e^{i \frac{\pi}{6}}} \frac{z-e^{i \frac{\pi}{6}}}{1+z^{6}}$

$\displaystyle r_{1} = \lim_{z \rightarrow e^{i \frac{\pi}{2}}} \frac{z-e^{i \frac{\pi}{2}}}{1+z^{6}}$

$\displaystyle r_{2} = \lim_{z \rightarrow e^{i \frac{5 \pi}{6}}} \frac{z-e^{i \frac{5 \pi}{6}}}{1+z^{6}}$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• May 28th 2010, 03:41 PM
fredrick08
Quote:

Originally Posted by chisigma
The poles of the function $\displaystyle f(z)= \frac{1}{1+z^{6}}$ are at $\displaystyle z= e^{\pm i \pi \frac{1+2k}{6}}$ , $\displaystyle k=0,1,2$. Only the residues of the poles with positive imaginary part give contribution of the integral that is...

$\displaystyle \int_{-\infty}^{+\infty} \frac{dx}{1+x^{6}} = 2 \pi i \sum_{k=0}^{2} r_{k}$ (1)

... where...

$\displaystyle r_{0} = \lim_{z \rightarrow e^{i \frac{\pi}{6}}} \frac{z-e^{i \frac{\pi}{6}}}{1+z^{6}}$

$\displaystyle r_{1} = \lim_{z \rightarrow e^{i \frac{\pi}{2}}} \frac{z-e^{i \frac{\pi}{2}}}{1+z^{6}}$

$\displaystyle r_{2} = \lim_{z \rightarrow e^{i \frac{5 \pi}{6}}} \frac{z-e^{i \frac{5 \pi}{6}}}{1+z^{6}}$ (2)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

srry im having a lot of trouble with these limits?? dont they all goto zero.. as z approaches to what equal in the top line????

srry dumb question.... Thankyou very much for ur help
• May 29th 2010, 01:54 AM
HallsofIvy
No, the limits are not all 0. If $\displaystyle z= e^{i\pi/6}$ the numerator is 0 but $\displaystyle z^6= e^{i\pi}= -1$ so the denominator, $\displaystyle z^6+ 1$ is also 0. Use L'Hopital's rule to find the limits.
• May 30th 2010, 07:27 PM
igopogo
I think there's a simpler way. Consider

$\displaystyle f(z)=\frac{p(z)}{q(z)}$ where

$\displaystyle p(z) = 1$ and

$\displaystyle q(z)=z^6+1$

Then $\displaystyle p(z_i) \neq 0$, $\displaystyle q(z_i)=0$ and $\displaystyle q'(z_i) \neq 0$ for each of your singularities.

Then the residue at $\displaystyle z_i$ is $\displaystyle \frac{p(z_i)}{q'(z_i)}=\frac{1}{6{z_i}^5}$

There are usually singularities in the UHP and LHP with these problems, but the LHP singularities are never inside the contour that you construct.