Surely a bug

For another story about mathematics and software bugs, see here:

I don’t know any interesting bugs in symbolic algebra packages but I know a true, enlightening and entertaining story about something that looked like a bug but wasn’t.$\def\sinc{\operatorname{sinc}}$

Define $\sinc x = (\sin x)/x$.

Someone found the following result in an algebra package:

$\int_0^\infty dx \sinc x = \pi/2$

They then found the following results:

$\int_0^\infty dx \sinc x \; \sinc (x/3)= \pi/2$

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5)= \pi/2$

and so on up to

$\int_0^\infty dx \sinc x \; \sinc (x/3) \; \sinc (x/5) \; \cdots \; \sinc (x/13)= \pi/2$

So of course when they got:

$\int_0^\infty dx \sinc x \; \sinc (x/3) \sinc (x/5) \; \cdots \; \sinc (x/15)$$= \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi$

they knew they had to report the bug. The poor vendor struggled for a long time trying to fix it but eventually came to the stunning realisation that this result is correct.

These are now known as Borwein Integrals.

by Dan Piponi answered Jan 13 ‘10 at 2:14 and comments:

The actual person at that “poor vendor” was me. I must have spent 3 days on this problem before I figured out that Jon had tricked me. And, indeed, I am an expert in computer algebra, but do not know much Fourier analysis. But Jon’s proof for why this is ‘correct’ is quite geometrical. – Jacques Carette Feb 17 ‘10 at 4:03

@Voyska No, it was not reported as a bug, just as an ‘oddity’ (or something like that). Jon was not mean, but playful in a devious way. He will be missed. – Jacques Carette Aug 18 ‘16 at 18:18