# Clarification of Cauchy Principal Value and use of Contour Integration

I am evaluating the improper integral $\int_{-\infty}^\infty{\frac{\sin^3 x}{x^3}dx}$; I am also told to show that this is equal to its principal value, and use this fact to evaluate the integral.

I have already evaluated the integral in the past by seeing this as the imaginary part of the integral $\int_{-\infty}^\infty{\frac{-e^{3ix}+3e^{ix}+2}{4x^3}dx}$ and solving this by using a dimpled semicircle of radius $R$ centered at the origin (with the dimple a semicircle of radius $\epsilon<R$ also centered at the origin). Using this methodology provides the correct result of $3\pi/4$.

Knowing about the Cauchy Principal value is making me re-evaluate the use of contour integration to evaluate real integrals. It seems to me that the above use of the dimpled semi-circle implicitly assumed that the integral was equal to its principal value since both sides of the integral on the real line were treated with the same values $R$ and $\epsilon$.

My questions seem to be concerned with the better/more acceptable approach to dealing with this issue, then: In justifying that the principal value of the integral is equal to the integral itself, should I

1. just realize that the integrand is analytic everywhere within the contour or everywhere but finitely-many points, at which point I can deform the contour in any way I see fit, meaning that I wouldn't need to have semicircles of different sizes on either side of the real line?
2. or should I first use the contour integration method to evaluate just a single side of the integral, and then the second side will follow from the fact that the integrand is even?

Personally, I feel the first method makes better use of the properties of the integrand, while the second is not as generalizable since for non-even integrands this would boil down to just carrying out the contour integration on both sides separately, which takes a little more work and doesn't really seem illuminating at all.

• Perhaps this might help: math.stackexchange.com/questions/406939/… Mar 1, 2014 at 16:07
• I'm not sure, as I haven't tried it, but couldn't this be solved with u-substitution where $U = x^3$? Mar 1, 2014 at 16:50
• I'm not interested in solving it; I have already done that. I just wanted to know which of the two explanations of the fact that the principal value of the integral is equal to the integral itself are considered better. Mar 1, 2014 at 19:02

The explanation I'd use is the following: we know that $\frac{\sin^3(x)}{x^3}$ is an integrable function through standard analysis (observe that the $\sin^3(x)$ term decays like $x^3$ near 0 whereas the $x^3$ in the denominator guarantees the convergence of the integral away from 0).
Now, for fixed $\epsilon > 0$, define $f_{\epsilon}(x) := \frac{\sin^3(x)}{x^3}$ for $|x| > \epsilon$ and $0$ otherwise, and define $f(x) := \frac{\sin^3(x)}{x^3}$. It should not be hard to show that $f_{\epsilon} \to f$ uniformly as $\epsilon \to 0^+$. So, given $R > 0$,
$\displaystyle \int_{\epsilon < |x| < R} \frac{\sin^3(x)}{x^3} dx \to \int_{|x| < R} \frac{\sin^3(x)}{x^3} dx$ as $\epsilon \to 0^+$.