Using residue theorem, calculate
$\displaystyle \int_{-\infty}^\infty \frac {1}{1+x^4} dx $
How do I find the poles of this function?
That is almost right, $\displaystyle x^4 =-1 ~ \Rightarrow x^2 = \pm \sqrt{-1} = \pm i$
What contour would you use to evaluate this integral? Which of the singularities lie within this contour and can you calculate the residues?
The four values are
$\displaystyle i^\frac{1}{4}, -i^\frac{1}{4}, i^\frac{-1}{4},-i^\frac{-1}{4} $
Since the integral limits are $\displaystyle -\infty$ and $\displaystyle \infty$
we take the positive part of the integral and use the positive values
which are
$\displaystyle i^\frac{1}{4}$ and $\displaystyle i^\frac{-1}{4}$
So
$\displaystyle Res(f,i^\frac{1}{4}) = \frac{1}{(x + i^\frac{1}{4})(x + i^\frac{-1}{4}), (x- i^\frac{-1}{4})}$
$\displaystyle = \frac{1}{(2i^\frac{1}{4})(i^\frac{1}{4} + i^\frac{-1}{4})(i^\frac{1}{4} - i^\frac{-1}{4})} $
so the integral will be $\displaystyle 2\pi i * Res(f,i^\frac{1}{4})$
=
I think somewhere is wrong.
Quite frankly, the fact that you keep writing "$\displaystyle i^{1/4}$" indicates you need to brush up on "complex numbers" before trying contour integrals and residues.
i can be written in "polar form" as $\displaystyle e^{i\pi/2}$. It's fourth roots are given by $\displaystyle e^{\frac{i(\pi/2+ 2\pi k)}{4}}$ which gives different result for k= 0, 1, 2, and 3.
With k= 0, a fourth root of i is $\displaystyle e^{i\pi/8}= cos(\pi/8)+ i sin(\pi/8)$. With k= 1, another is $\displaystyle e^{5i\pi/8}= cos(5\pi/8)+ i sin(5\pi/8)$. With k= 2, $\displaystyle e^{9i\pi/8}= cos(9\pi/8)+ i sin(9\pi/8)$. Finally, with k= 3, $\displaystyle e^{13\pi/8}= cos(13\pi/8)+ i sin(13\pi/8)$.
Two of those four roots have positive real part, two negative real part. Two have positive imaginary part, two negative imaginary part. That means you really only need to look at the residues at two poles, depending on how you set up the contour.
$\displaystyle x^{4} +1 = 0 $
$\displaystyle x^{4} = -1 $
$\displaystyle x^{4} = e^{i \pi} $
$\displaystyle x = e^{ \frac{i( \pi + 2 \pi k)}{4}} $ for k = 0, 1, 2, and 3
so the roots are $\displaystyle e^{\frac{i \pi}{4}}, e^{\frac{3 i \pi}{4}}, e^{\frac{5 i \pi}{4}}, e^{\frac{7 i \pi}{4}} $
For the countour integration you only need the two poles in the upper half of complex plane ($\displaystyle e^{\frac{i \pi}{4}}$ and $\displaystyle e^{\frac{3 i \pi}{4}} $)
$\displaystyle e^{\frac{i \pi}{4}} = \cos(\pi /4) + i \sin(\pi /4) = \sqrt{2}/2+ i \sqrt{2}/2$
$\displaystyle ^{\frac{i 3\ pi}{4}} = \cos(3 \pi /4) + i \sin(3 \pi /4) = -\sqrt{2}/2+ i \sqrt{2}/2$
$\displaystyle \int^{\infty}_{\text{-}\infty} \frac {1}{1+x^{4}} $
let $\displaystyle f(z) = \frac {1}{1+z^4} $
As I mentioned, there are two simple poles in the upper half of the complex plane.
$\displaystyle Res \{f, z_{o}\} = \lim_{z \to z_{0}} (z-z_{0}) \frac {1}{1+z^{4}} = \lim_{z \to z_{0}} \frac {1}{4z^{3}} = \frac {1}{4z_{0}^{3}}$
$\displaystyle Res \{f, \sqrt{2}/2+ i \sqrt{2}/2\} = \text{-} \sqrt{2} /8 - i \sqrt{2} /8 $ (after simplification)
$\displaystyle Res \{f, - \sqrt{2}/2+ i \sqrt{2}/2\} = \sqrt{2} /8 - i \sqrt{2} /8 $
$\displaystyle \int^{\infty}_{\text{-}\infty} \frac {1}{1+x^{4}} = 2 \pi i \sum (\text{residues in the upper half plane})$ $\displaystyle = 2 \pi i (\text{-}i \sqrt{2} /4 ) = \pi \frac{\sqrt{2}}{2}$