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I am doing some exercises in complex integration. I am given the two functions:

$$\frac{\sqrt{x}}{x^2+4}$$

$$\frac{1}{\sqrt{x}(x^2+1)}$$

They must be integrated from $0$ to $+\infty$ with a complex integration. From the solution I find that the correct path of integration for the first function is:

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while for the second is:

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So in the first integration both poles of the function contribute to the integral, but in the second only the pole at $+i$ contributes. I am lost. The two functions seem similar, why this difference?

On a more general ground: are there criteria for picking the correct integration path?

mattiav27
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1 Answers1

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I think that, in both cases, the easiest way is to perform the substitution $x=y^2$ and $\mathrm dx=2y\,\mathrm dy$. The first integral now becomes$$2\int_0^\infty\frac{y^2}{y^4+4}\,\mathrm dy=\int_{-\infty}^\infty\frac{y^2}{y^4+4}\,\mathrm dy.$$And now you use the standard path for these situations:

enter image description here

That is, you go from $-R$ to $R$ along the real numbers and then you go from $R$ to $-R$ along a half-circle in the upper half-plane. You shall get that your integral is equal to $\frac\pi2$.

In the case of the second integral, you shall get$$\int_{-\infty}^\infty\frac1{y^4+1}\,\mathrm dy$$and the rest is similar.

José Carlos Santos
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