Calculate the following integral using Residue Theorem
$\displaystyle \int \frac{dz}{(1+z^2)^2}$
from $\displaystyle \infty$ to $\displaystyle -\infty$
Hello,
Let $\displaystyle f(z)=\frac{1}{(1+z^2)^2}$
Obviously, i and -i are double poles, because $\displaystyle 1+z^2=(z+i)(z-i)$
So, using the general formula for a pole of order n : $\displaystyle \text{Res}_a f(z)=\frac{1}{(n-1)!} \lim_{z\to a}\frac{\partial^{n-1}}{\partial z^{n-1}} (z-a)^n f(z)$
we easily get the residue of f at i (the one that will be useful for the computation) :
$\displaystyle \text{Res}_{z=i} f(z)=\lim_{z\to i} \frac{d}{dz} (z-i)^2 f(z)=\lim_{z\to i} \frac{d}{dz} \frac{1}{(z+i)^2}$
which is easy to compute.
Spoiler:
Now, use the semicircle contour :
And keep the poles with positive imaginary part. So only i.
And then $\displaystyle \int_{-\infty}^\infty f(z) ~dz=2i\pi \text{Res}_{z=i} f(z)$
ok. so for
$\displaystyle Res(f,i) = \lim_{z\to i} \frac{d}{dz}(z-i)^2f(z) $
$\displaystyle = \lim_{z\to i} \frac{d}{dz} \frac {1}{z+i}^2$
$\displaystyle =\lim_{z\to i} \frac{-(2z+2i)}{(z+i)^4}$
$\displaystyle = \frac {-4i}{16i^4}
= \frac{i}{4} $
and
$\displaystyle Res(f,-i) = \lim_{z\to -i} \frac{d}{dz}(z+i)^2f(z) $
$\displaystyle = \lim_{z\to -i} \frac{d}{dz} \frac {1}{z-i}^2$
$\displaystyle =\lim_{z\to -i} \frac{-(2z-2i)}{(z-i)^4}$
$\displaystyle = \frac {4i}{16i^4}
= \frac {-i}{4} $
Am I right to say that you took semi contour and the positive side of the imaginary because of the -infinity to infinity?
The last part would be
$\displaystyle \int_{-\infty}^\infty f(z) ~dz=2\pi i Res(f,i)$
$\displaystyle = 2 \pi i * \frac {i}{4} = \frac {-\pi}{2}$
Hey...
$\displaystyle i^4=(i^2)^2=(-1)^2=1$
So $\displaystyle = \frac {-4i}{16i^4}
= {\color{red}-}\frac{i}{4}
$
You got wrong for the second one too.
Isn't it surprising to get a negative answer for the integral of a positive function ?
Yes. If we had the integral from infinity to -infinity, it would have been more logic to take the semi circle and the negative side of the imaginary.Am I right to say that you took semi contour and the positive side of the imaginary because of the -infinity to infinity?
But the result is actually the same since $\displaystyle \int_{-\infty}^\infty =-\int_\infty^{-\infty}$
But more commonly, we take the upper semi circle
I hope I'm clear :s