Wrong complex integral

Question

I've tried to perform the following integral with mathematica (student edition):

Integrate[(I/Pi)/(-x + I/(2 t)), {x, -Infinity, Infinity}, Assumptions -> t> 0]

Mathematica says that this integral doesn't converge but it is wrong, the right result is 1. Is something wrong with the string?

Answer 1

This integral exist only in the principal value sense:

Integrate[(I/Pi)/(-x + I/(2 t)), {x, -Infinity, Infinity},
 Assumptions -> t > 0, PrincipalValue -> True]

1

That means, the integrand is not in $L^1(\mathbb{R};\mathbb{R})$, but still

$$\lim_{R \to \infty} \int_{-R}^R \frac{\operatorname{i}}{\pi \left(-x+\frac{\operatorname{i}}{2 t}\right)} \, \operatorname{d} x = 1$$

exists. The important point is that

$$ \int_{-S}^R \frac{\operatorname{i}}{\pi \left(-x+\frac{\operatorname{i}}{2 t}\right)} \, \operatorname{d} x = 1$$

only converges if the lower and upper integral boundary $S$ and $R$ converge to $\infty$ with (essentially) same rate. The reason is that the imaginary part of the integrand is

Im[(I/Pi)/(-x + I/(2 t))] // ComplexExpand // TeXForm

$$-\frac{x}{\pi \left(\frac{1}{4 t^2}+x^2\right)},$$

so it is an odd function in $x$ and goes assymptotically like $\frac{1}{x}$ which is not (absolutely) integrable.