How to know which contour to use?

Question

I'm currently taking a class on complex analysis, and I'm not sure why the professor uses certain contours to integrate. For example, when trying to integrate $$ \int_{-\infty}^\infty \frac{\sin x}{x(x^2+4)} dx $$ they integrate over the following contour 1

Then, when integrating $$ \int_{0}^\infty \frac{x^a}{x^2+3x+2} dx, 0 < a< 1 $$ they use the keyhole contour. I understand the two functions are different, but I don't understand how or why we integrate over the different contours and not just the usual upper-half circle. Futhermore, what is stopping us from integrating over the upper-half circle alone?

Answer 1

For the integral we typically first rewrite $$\int_{-\infty}^\infty \frac{\sin x\,dx}{x(x^2 + 4)} = \Im \int_{-\infty}^\infty \frac{e^{iz} \,dz}{z (z^2 + 4)}$$ (what happens if we try to evaluate $\int_{-\infty}^\infty \frac{\sin z\,dz}{z(z^2 + 4)}$ using a semicircular contour?). The integrand has a pole at $z = 0$, hence to apply the Residue Theorem to evaluate a contour integral $\oint_\Gamma \frac{e^{iz} \,dz}{z (z^2 + 1)}$, the contour $\Gamma$ must not pass through $0$. Thus, we take a semicircular detour of radius $\epsilon$ about $0$ and compute the limit as $\epsilon \to 0$.

In the second case, $$\int_{0}^\infty \frac{x^a}{x^2+3x+2} dx$$ integrating over a semicircular path poses three problems:

1. The integrand has poles on the negative half of the real axis (at $x = -1$ and $x = -2$), so we encourter the same difficulty as for the first integral.
2. The function $x^a$ has a branch cut that includes $0$, so any contour $\Gamma$ to which we apply the Residue Theorem must avoid $0$.
3. Even if we were evaluating, e.g., the similar integral $$\int_0^\infty \frac{x^a \,dx}{x^2 + 2 x + 2}$$ and addressed the issue in (2) by using a contour like the one used for the first integral, we run into another issue: (Supposing we take the branch cut characterized by $\arg z \in [0, 2\pi)$), the integral of $$\frac{z^a \,dx}{z^2 + 2 z + 2}$$ over the straight-line contour $L$ from $-R$ to $-\epsilon$, $0 < \epsilon < R$, is $$\int_L \frac{z^a \,dz}{z^2 + 2 z + 2} = -\int_\epsilon^R \frac{x^a \,dx}{x^2 - 2 x + 2},$$ which is not a multiple of the integral $$\int_\epsilon^R \frac{x^a \,dx}{x^2 + 2 x + 2}$$ we're aiming to compute. This issue would not occur if the denominator of the original integral were, e.g., an even polynomial in $x$.