I’m trying to understand how to solve the following integral:
$$\int_0^{\infty}\frac{x^{1/4}}{x^2+1}dx$$
From my understanding I need to draw a Branch cut but I’m having troubles understanding where to do it in order to set up my contour to integrate using the residue theorem.
I understand how it works for a normal square root but here I’m a bit lost.
This is what I’ve tried:
The function has simple poles at $z=\pm i$ and this is the contour I tried:
$$\Gamma_1 = \big\{z=Re^{i\theta}, 0<\theta<2\pi, R>0\big\}$$
$$\Gamma_2= \big\{z=re^{i2\pi}, \epsilon<r<R\big\}$$
$$\Gamma_3 = \big\{z=\epsilon e^{i\theta}, 0<\theta<2\pi\big\}$$
$$\Gamma_4 = \big\{z=re^{i0}, \epsilon<r<R\big\}$$
So a “keyhole” contour as my professor says.
Due to the estimation Lemma and the fact that the function is regular, we ignore the contributions of $\Gamma_{1,3}$
On $\Gamma_2$ we end up with $-iI$ where $I$ is the integral.
On $\Gamma_1$ we end up with $I$ and using the Residue theorem:
$$I(1-i) = 2\pi \cos(\pi/8)$$.
This seems to not be the actual answer and I don’t know where I went wrong.
Many thanks!