Evaluate :$$ \int_{0}^{\infty} \frac{x^4}{(1+2x^2)^4} dx$$
using residue theorem. The integral is even and it can be written as :
$$ \frac{1}{2} \int_{-\infty}^{\infty} \frac{x^4}{(1+2x^2)^4} dx$$
The only pole with a positive imaginary part is $ \frac{i}{\sqrt2} $ and its degree is $4$. How can I evaluate this without differentiating 3 times when calculating the residue?
I also tried calculating the infinite residue but that didn't help much.
Best Answer
You can instead find the laurent series, and pick up the coefficient of the $z^{-1}$ term. i.e. $(1+2z^2)^{-4} = (1+2(z-\frac{i}{\sqrt2})^2-2(\frac{-1}{2})+4\frac{iz}{\sqrt2})^{-4}= (1+2(z-\frac{i}{\sqrt2})^2+1+2\sqrt2iz)^{-4} = (2\sqrt{2}i(z-\frac{i}{\sqrt2})+2\sqrt2i\frac{i}{\sqrt2}+2+2(z-\frac{i}{\sqrt2})^2)^{-4}=(2\sqrt2i(z-\frac{i}{\sqrt2})+2(z-\frac{i}{\sqrt2})^2)^{-4}$
Put $y=z-\frac{i}{\sqrt2}$, so you need to find the laurent series of $(2\sqrt{2}iy+2y^2)^{-4}=(2\sqrt2iy)^{-4}(1+\frac{-i}{\sqrt2}y)^{-4}$.
Writing $\frac{z^4}{(1+2z^2)^{4}}=\frac{(y+\frac{i}{\sqrt2})^4}{(2\sqrt2iy)^{4}(1+\frac{-i}{\sqrt2}y)^{4}}=\frac{(y+\frac{i}{\sqrt2})^4}{64y^4(1+\frac{-i}{\sqrt2}y)^{4}}$
Therefore you are only interested in the $y^{-1}$ coefficient which corresponds to $y^3$ coefficient of $\frac{(y+\frac{i}{\sqrt2})^4}{(1+\frac{-i}{\sqrt2}y)^{4}}$ scaled by $\frac{1}{64}$.
Looking at $(y+\frac{i}{\sqrt2})^4(1+\frac{-i}{\sqrt2}y)^{-4}=(1/4 - i \sqrt2 y - 3 y^2 + 2 i \sqrt2 y^3 + y^4)(1+2\sqrt2iy-5y^2-5\sqrt2iy^3+O(y^4))=....-\frac{i}{2\sqrt2}y^3+....$
Scaling that by $\frac{1}{64}$, we find the residue to be $\frac{-i}{128\sqrt2}$