## 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!

Labels: 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.

Thursday, 02 March, 2006 Unknown said...

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

Thanks.

Wednesday, 08 March, 2006
{"ft_lang_label":"__label__en","ft_lang_prob":0.8979268,"math_prob":0.9647207,"size":2878,"snap":"2020-10-2020-16","text_gpt3_token_len":941,"char_repetition_ratio":0.22790536,"word_repetition_ratio":0.79301745,"special_character_ratio":0.33391243,"punctuation_ratio":0.08435583,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9693148,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-23T07:55:10Z\",\"WARC-Record-ID\":\"<urn:uuid:7fd5b70d-80a2-44a8-a2a0-bca62cd44f48>\",\"Content-Length\":\"17948\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:acb8ee42-89ac-4be7-b61d-2ce7235a7933>\",\"WARC-Concurrent-To\":\"<urn:uuid:6a5f2351-813a-4529-b27d-7593fcbf6744>\",\"WARC-IP-Address\":\"172.217.7.179\",\"WARC-Target-URI\":\"http://blog.sigfpe.com/2006/03/cautionary-tale-for-would-be.html\",\"WARC-Payload-Digest\":\"sha1:D4CVA4HUTRCVSUWZHFIQ2JHN553JUZDZ\",\"WARC-Block-Digest\":\"sha1:ZPOYWBI3Y5MQJ44BUVI6T653WCA6JSAL\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875145747.6_warc_CC-MAIN-20200223062700-20200223092700-00283.warc.gz\"}"}