How I Impressed My Wife: Part 1

Some husbands try to impress their wives by lifting extremely heavy objects or other extraordinary feats of physical prowess.

That will never happen in the Quintanilla household in a million years.

But she was impressed that I broke an impasse in her research and resolved a discrepancy between Mathematica 4 and Mathematica 8 by finding the following integral by hand in less than an hour:

\displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}

Yes, I married well indeed.

This integral serves as the theoretical underpinnings for finding vortices in the velocity fields of atomic wave functions and can be obtained from Equation (4) of a 2014 paper in Physical Review A that she co-authored. And I really like this integral because there are so many different ways of evaluating it, including various trigonometric identities, the magic substitution u = \tan (x/2), partial fractions, and even contour integration and residues.

In this series, I’ll explore different ways of evaluating this integral, starting tomorrow. Until then, I offer a thought bubble if you’d like to try to tackle this before I present the solution(s).


2 thoughts on “How I Impressed My Wife: Part 1

  1. Reblogged this on nebusresearch and commented:
    I’m embarrassed to come to this late. I can only plead that it’s been rather busy around my place the last few weeks and I haven’t been doing as much reading of other people’s blogs as I ought. Anyway. John Quintanilla begins here a series about evaluating a particular and somewhat ugly-looking integral. His wife needed the result and Mathematica wasn’t of help. Or, worse, really: Mathematica 4 and Mathematica 8 didn’t agree on what the integral should be. With the computer’s limits found John Quintanilla turned to reasoning, the kind you do when you learn calculus well.
    Quintanilla found multiple ways to evaluate this integral, and has been spending several weeks sharing them. I recommend the series. I admit it probably won’t mean much if you haven’t taken calculus. It also won’t be enlightening if you don’t remember much about substitutions. But if you remember how to do integrals at all you surely remember how to do substitutions. At least you’ll get the rust out of them soon enough.

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