Wednesday, March 01, 2006

A Cautionary Tale for Would-Be Generalisers

You may or may not know this story already as it's been floating around for a while. But I'm going to retell it anyway:

Define sinc(x) = sin(x)/x and sinc(0) = 1

Let 'I' mean integration from 0 to infinity.

I sinc(x) = pi/2
I sinc(x)sinc(x/3) = pi/2
I sinc(x)sinc(x/3)sinc(x/5) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9) = pi/2

You see the pattern right? So the story is that some guy was evaluating these integrals for some reason or other getting pi/2 all the time. They happily chugged on getting...

I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11) = pi/2
I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11)sinc(x/13) = pi/2

So when the computer algebra package this person was using said:

I sinc(x)sinc(x/3)sinc(x/5)sinc(x/7)sinc(x/9)sinc(x/11)sinc(x/13)sinc(x/15) = 467807924713440738696537864469/935615849440640907310521750000*pi

they knew they'd hit a bug. They complained to the vendor who agreed that, yes, definitely, there was something screwy in their integration routine.

Except there wasn't. Weird eh?

I've known this story for a while but I've only just stumbled on Borwein's paper that explains the phenomenon. They're called Borwein Integrals now.

Update: Corrected a small error. It's pi/2 for the first 7 terms, not just the first 6. Even more amazing!

2 comments:

David R. MacIver said...

I don't really have anything useful to contribute, except to say that that's really cool. :-)

Thanks for sharing - I'd not seen it before.

P.P. Cook said...

Where mr maciver leads, i wish to follow, and say: that is really interesting.

Thanks.

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