MORPHING LORD BROUNCKER S CONTINUED FRACTION FOR PI INTO THE PRODUCT OF WALLIS

Transcription

1 MORPHING ORD BROUNCKER S CONTINUED FRACTION FOR PI INTO THE PRODUCT OF WAIS Thomas J. Osler Mathematics Departmet Rowa Uiversity Glassboro, NJ 00 osler@rowa.edu Itroductio Three of the oldest ad most celebrated formulas for pi are: (), (), ad (). The first is Vieta s product of ested radicals from []. The secod is Wallis s product of ratioal umbers [] from ad the third is ord Broucker s cotiued fractio [,], also from. (I the remaider of the paper we will use the more coveiet otatio for cotiued fractios.) I a previous paper [] the author showed that () ad () are actually special cases of a more geeral formula

2 () k k k k. k ( k radicals) k By eamiig this formula for the sequece of special values 0,,,, we observe that the product of Wallis (case 0) appears to gradually morph ito Vieta s product as approaches ifiity. We illustrate this below: 0: 0 0 (origial Wallis s product) : 0 0 : : : (origial Vieta s product) Observe that as we progress through each step of the sequece, oe additioal factor of Vieta s product is added, while every other fractio, startig with the first, i the Wallis type product is removed. We will show that Broucker s cotiued fractio () ad the product of Wallis () are both special cases of the geeral formula () W ( ) ( ) ( ) ( ) ( ),

3 i which () W ) )( ( ) ( is the partial Wallis product. Just as above, this geeral formula allows us to start with ord Broucker s cotiued fractio (case 0 ) ad gradually morph it ito the Wallis product as approaches ifiity. 0 : (ord Broucker s cotiued fractio.) : : : : (origial product of Wallis) Observe that as we progress through each step of the sequece, oe additioal factor of Wallis s product is added, while the Broucker type cotiued fractio has the value of icremeted by. I a recet paper [] age called attetio to a cotiued fractio for pi resemblig Broucker s fractio ()

4 (). We show that the geeral formula () ( ), W ( ) ( ) ( ) ( ) ( ) like () cotais age s cotiued fractio () ad the product of Wallis () as special cases. Agai by eamiig this formula as show below: 0,,,, we ca morph () ito () as 0:. (age s cotiued fractio.) :. :. : :. (Origial product of Wallis reciprocated.) Etesios of the formulas () ad () are also give. Derivatio of the results ad more morphig All the results of this paper are special cases of the kow formula [, page ]

5 y y Γ Γ () y y y, y y Γ Γ valid for either y a odd iteger ad ay comple umber or y ay comple umber ad Re( ) > 0. The ames of Euler, Stieltes, ad Ramaua [, page 0] have bee associated with this result. Usig the very well kow formulas Γ ( )!, Γ( ) Γ( ) ad k (k ) Γ ( / ) we have Γ, valid for k k,,,. With this last result ad appropriate values of ad y, the left had side of () ca be epressed i terms of ratioal umbers ad. For eample, if we set y 0 ad i () we get our geeral formula () ad settig y 0 ad we get our geeral formula (). The maipulatios are simple ad the reader will have o difficulty verifyig our formulas. If i () we set ad, y ( ) for a iteger i the rage 0 we get a etesio of () (0) ( )( ) ( ) W ( ( )( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ). et 0 ad (0) becomes ( ) W ( ) ( )). ( ) ( ) ( )

6 We list the special cases of this formula for, 0,, below: 0: : : : : (origial Wallis product) If i () we set ad ) ( y for a iteger i the rage we get a etesio of () < () ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) )( ( ) ( ) )( ( W. et ad () becomes ) ( ) ( ) ( ) ( ) ( ) )( ( ) )( ( W. et us list this formula for various values o :

7 : : : 0 : 0 : (origial Wallis product reciprocated) We see that the geeral formulas (0) ad () start with a geeralized Broucker type cotiued fractio y y y ad gradually morph it ito the Wallis product as approaches ifiity. Fial remarks Wallis described the igeious way i which he obtaied his product () i []. He states that he showed his product to ord Broucker who the obtaied the cotiued fractio (). It appears that Broucker ever published his method of fidig this cotiued fractio ad oly partially eplaied his reasoig to Wallis. Wallis gives some hits i [, pages - ] as to how Broucker proceeded but the eplaatio is icomplete. Stedall i [, pages 00-0] has discussed this ad made her ow coecture as to how Broucker might have reasoed. I his discussio of this questio, Wallis published a table i [, page ] which we reproduce here.

8 I the third row of this table we see the cotiued fractios obtaied from our geeral formulas () ad (). Stedall [] has recetly called attetio to these fractios that appear to have bee overlooked. She also poits out [, page 0] that both Wallis ad Broucker could easily have writte the value of these fractios i terms of ratioal umbers ad pi. Thus we see that the cotiued fractios that we obtaied from () ad () are amog the oldest cotiued fractios ad their values were coectured as early as! Ackowledgemet The author wishes to thak James Smoak for his geerous assistace with the historical items i this paper. Refereces [] Berdt, B. C., Ramaua s Notebooks, Part II, Spriger-Verlag, New York,. [] age,. J., A Elegat Cotiued Fractio for, The America Mathematical Mothly, 0 (), pp. -. [] Osler, T. J., The uited Vieta s ad Wallis s products for pi, America Mathematical Mothly, 0 (), pp. -.

9 [] Perro, O., Die ehre vo de Kettebruche, Bad II, Teuber, Stuttgart,. [] Stedall, Jacquelie A., Catchig Proteus: The Collaboratios of Wallis ad Broucker. I. Squarig the Circle, Notes ad Records of the Royal Society of odo, Vol., No., (Sep., 000), pp. - [] Vieta, F., Variorum de Rebus Mathematicis Reposorum iber VII, () i: Opera Mathematica, (reprited) Georg Olms Verlag, Hildesheim, New York, 0, pp. -00 ad -. [] Wallis, Joh, The Arithmetic of Ifiitesimals, (Traslated from ati by Jacquelie A. Stedall), Spriger Verlag, New York, 00.