A collection of mathematical formulas involving π

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1 A collectio of mathematical formulas ivolvig David H. Bailey February 6, 8 Abstract This ote presets a collectio of mathematical formulas ivolvig the mathematical costat. Backgroud The mathematical costat kow as is udeiably the most famous ad arguably the most importat mathematical costat. Mathematicias sice the days of Euclid ad Archimedes, up to ad icludig the preset day, have aalyzed its properties ad computed its umerical value. This is a collectio of may of formulas that have bee established by mathematicias over the years ivolvig. While a comprehesive collectio is of course ot possible, preferece is give i this list for formulas that satisfy the followig criteria: Formulas that give (or a very simple expressio ivolvig explicitly, as opposed to implicit relatios such as e i +. Formulas that give (or a very simple expressio ivolvig as a straightforward ifiite series, ifiite product or defiite itegral. Formulas that ivolve oly simple otatio, such as summatios, itegrals, biomial coefficiets, square roots, expoetials, logarithms, etc., that would be familiar to ayoe who has completed a begiig course i calculus. Iterative algorithms for ivolvig simple expressios of the above types. Formulas that are relatively ew, discovered withi the last 5 years or so. Icluded i this listig are several formulas for that have actually have bee used i large calculatios of, both before ad sice the ivetio of the computer. These iclude formulas through 5 prior to the th cetury, ad formulas 6, 7,,, 3,, 6, 8, 75 ad 76 i the late th ad early st cetury. Formulas 3 through 8 have the itriguig property that they permit digits (i certai specific bases of the costat specified o the left-had side to be calculated begiig at a arbitrary startig positio, without havig to calculate ay of the digits that came before, by meas of a relatively simple algorithm. Formulas 3 ad have bee used i computatios of high-order biary digits of [9, Sec ], while formula 6 has bee used i computatios of high-order biary digits of, ad formula 8 has bee used i computatios of high-order base-3 digits of [6]. Numerous similar recetly-discovered formulas that possess the arbitrary digit-computatio property for various mathematical costats are catalogued i []. Lawrece Berkeley Natioal Laboratory, Berkeley, CA 97 (retired ad Uiversity of Califoria, Davis, Departmet of Computer Sciece, dhbailey@lbl.gov.

2 May of these formulas are relatively ew, i the sese that they were discovered oly i the past 5 years or so. The formulas metioed i the previous paragraph are certaily i this category, havig bee discovered oly sice 996. May of the formulas from 9 through 5 were ot well kow util recetly. Formulas 68 through 7 are also relatively ew, i the sese that they are part of a class of itegral formulas that are the subject of curret research [3,, 5]. Formula 75 was discovered i 976. Formulas 76 ad 77 were discovered i 98. Credits Formula was discovered by Leibiz ad Gregory i the 6s. Formula was attributed to Euler i 738. Formula 3 was discovered about the same time by Machi [9, pg. 5]. The related arctagetbased formulas, 5, 6 ad 7 were used by Dase, Ferguso, Kaada ad Kaada, respectively [9, pg. 6, 7, ]. Formula 8 is due to the Idia mathematicia Madhava of Sagamagramma, who lived i the late 3s ad early s [9, pg. 7]. Formula 9 was discovered by Newto i the mid-6s [9, pg. 6]. Formula was discovered by Wallis at about the same time. Formula is due to Ramauja, ad was used by Gosper i 986 to compute to over 7 millio digits. The similar but more complicated Formula is due to David ad Gregory Chudovsky, ad was used by them to compute to over oe billio decimal digits [9, pg. 8]. Formula 3 is kow as the BBP formula for, amed for the iitials of the co-authors of the 997 paper where it was first preseted [7][9, pg. 9 ]. It was discovered by a computer program ruig the PSLQ algorithm of mathematicia-sculptor Helama Ferguso [, 8]. Formula is a variat of the BBP formula due to Bellard [9, pg. ]. Formula 5 was foud by Helama Ferguso ad idepedetly by Adamchik ad Wago [6]. Formula 6 appeared i [7]. Formulas 7 ad 8 are due to David Broadhurst []. Some of the summatio formulas ivolvig factorials ad combiatorial coefficiets (i.e., formulas 9 through 5 were foud by Ramauja; others are due to David ad Gregory Chudovsky. The Chudovskys had these ad may other formulas of this geeral type iscribed o the floor of their research ceter at Brookly Polytechic Uiversity i New York City [3]. Four exceptios are Formula 37, which is due to Ramauja but appeared i [, pg. 88], Formulas 6 ad 7, which are due to Guillera [5], ad 5, which is due to Almkvist ad Guillera []. Formulas 53 through 69 have bee kow for may years; may are from [, pg. 5, 8, 3 3]. Formula 7 is a example of umerous formulas of this geeral type recetly discovered by computatioal methods, typically ivolvig the PSLQ algorithm [, 8], i studies of Isig theory i mathematical physics [3]. Formulas 7, 7 ad 73 are examples of recet discoveries, also by computatioal methods ivolvig the PSLQ algorithm, i the theory of box itegrals [, 5]. Formula 7, for istace, ca be thought of as specifyig the average distace from the origi to a poit i the uit 3-cube. Formula 7, the first of the iterative formulas, is mathematically equivalet to Archimedes approach ivolvig computig the areas of iscribed ad circumscribed regular polygos [9, pg. ]. Archimedes polygo approach was used for almost all computatios of i aciet times, icludig by the fifth cetury Chiese mathematicia Tsu Chug-Chih ad, evidetly, by the fifth cetury Idia mathematicia Aryabhata. Formula 75 is the Bret-Salami formula, the first quadratically coverget formula. It was discovered idepedetly by Richard Bret ad Eugee Salami i 976 [9, pg. 9 ]. Formula 76 (a cubically coverget iteratio ad 77 (a quartically coverget iteratio are due to Joatha ad Peter Borwei [9, pg. ].

3 3 Formulas for ( + ( ( ( ( ( ( ( + 3 ( ( ( (3 ( ( ( ( ( ( ( ( ( (5 + 8 ( ( ( (6 ( ( + 3( + (9 (!( ( 98 (! 396 (6!( ( (3!(! / ( ( ( + ( ( + 3 ( (5 + 3 (7 (8 ( 3

4 9 ( (6 + (6 + 8 ( (6 + + ( ( ( ( + 5 ( + 8 ( + 7 ( ( ( ( ( + + ( log ( ( ( ( ( 6 3 ( ( 3( ( 8 6 ( ( 3 ( 6 6 ( log 3 3 ( 3 (5 (5 6 ( 3 (6 ( (! (3!( (6!! (7 5 8 (! (3!( (6!! (8 ( 7 (! (3!( (6!! (9 ( /3 (! (3!( (6!! (3 (6 (7 (8 (9 ( ( (3

5 ( / ( log 58 ( ( 5 9 ( ( (3 8 (3 6 5 ( ( log log ( (!( + 3 (! + 3 ( ( (3 (!(6 + 3 (! 8 + (9/8 ( ( (35 (9/8 ( ( (36 (!(6 + 3 (! 88 + ( (!(8 + (! 3 + (!( + (! 9 + (!(8 + 3 (! 3 + (3 (!(6 + (! ( (!( + 3 (! 9 + ( ( ( ( 5( (7 (37 (38 (39 ( ( (5 (6 5

6 6 ( ( ( ( dx + x ( 8 (6!( (! (5 x dx 7 x ta x dx + log x ( x dx + x 8 x dx ( + x x ( + log 8 log x dx ex log / 3 + log log log x dx si x / log x dx x + x + log x ta x dx (56 xe x e x dx (59 log (cos x dx (6 log( + x dx x log( + x 3 dx x + x dx dy xy (8 (9 (5 (5 (53 (5 (55 (57 (58 (6 (6 (63 (6 (65 (66 6

7 Γ(/ x / e x dx (67 8 dx dy x y 6 + log( log + 6 log + 3 log( log( (68 dx dy + x + y ( log( + 3 Iteratios for ( x x + ( y y + ( xy dx dy (7 xy + dx dy dz x + y + z (7 (The Archimedes iteratio. Set a 3 ad b 3. Iterate x + y + z dx dy dz (7 a + (x + y + z 3/ dx dy dz (73 ab a + b, b + a +b. (7 The both a ad b coverge to : each iteratio decreases the distace betwee a ad b (which iterval cotais by a factor of approximately four. (The Bret-Salami iteratio. Set a, b / ad s /. Iterate a k a k + b k, b k a k b k, c k a k b k, s k s k k c k, p k a k s k. (75 The p k coverges quadratically to : each iteratio approximately doubles the umber of correct digits. (The Borwei cubic iteratio. Set a /3 ad s ( 3 /. Iterate 3 r k+ + ( s, s 3 k /3 k+ r k+, a k+ r k+a k 3 k (r k+. (76 The /a k coverges cubically to : each iteratio approximately triples the umber of correct digits. (The Borwei quartic iteratio. Set a 6 ad y. Iterate y k+ ( y k / + ( y k / a k+ a k ( + y k+ k+3 ( + y k+ + y k+. (77 The /a k coverges quartically to /: each iteratio approximately quadruples the umber of correct digits. 7

8 Refereces [] Gert Almkvist ad Jesus Guillera, Ramauja-like series for / ad strig theory, 7 Sept., available at [] David H. Bailey, A compedium of BBP-type formulas for mathematical costats, updated 9 April 3, available at [3] David H. Bailey, Joatha M. Borwei ad Richard E. Cradall, Itegrals of the Isig class, Joural of Physics A: Mathematical ad Geeral, vol. 39 (6, pg [] David H. Bailey, Joatha M. Borwei ad Richard E. Cradall, Box itegrals, Joural of Computatioal ad Applied Mathematics, vol. 6 (7, pg [5] David H. Bailey, Joatha M. Borwei ad Richard E. Cradall, Advaces i the theory of box itegrals, Mathematics of Computatio, vol. 79, o. 7 (Jul., pg [6] David H. Bailey, Joatha M. Borwei, Adrew Mattigly ad Gle Wightwick, The computatio of previously iaccessible digits of ad Catala s costat, Notices of the America Mathematical Society, vol. 6 (3, o. 7, pg [7] David H. Bailey, Peter B. Borwei ad Simo Plouffe, O the rapid computatio of various polylogarithmic costats, Mathematics of Computatio, vol. 66, o. 8 (Apr. 997, pg [8] David H. Bailey ad David J. Broadhurst, Parallel iteger relatio detectio: Techiques ad applicatios, Mathematics of Computatio, vol. 7, o. 36 (Oct., pg [9] Joatha M. Borwei ad David H. Bailey, Mathematics by Experimet: Plausible Reasoig i the st Cetury, AK Peters, Natick, MA, 8. [] Joatha M. Borwei, David H. Bailey ad Rolad Girgesoh, Experimetatio i Mathematics: Computatioal Paths to Discovery, AK Peters, Natick, MA. [] Joatha M. Borwei ad Peter B. Borwei, Pi ad the AGM: A Study i Aalytic Number Theory ad Computatioal Complexity, CMS Series of moographs ad Advaced Books i Mathematics, Joh Wiley, Hoboke, NJ, 987. [] David J. Broadhurst, Massive 3-loop Feyma diagrams reducible to SC* primitives of algebras of the sixth root of uity, mauscript, 998, available at [3] David ad Gregory Chudovsky, Listig of Ramauja-type formulas, copy i author s possessio,. [] Helama R. P. Ferguso, David H. Bailey ad Stephe Aro, Aalysis of PSLQ, a iteger relatio fidig algorithm, Mathematics of Computatio, vol. 68, o. 5 (Ja. 999, pg [5] Jesus Guillera, Some biomial series obtaied by the WZ-method, Advaces i Applied Mathematics, vol. 9 (, pg [6] Victor Adamchik ad Sta Wago, A simple formula for pi, America Mathematical Mothly, vol. (Nov. 997, pg

SOME NEW IDENTITIES INVOLVING π,

SOME NEW IDENTITIES INVOLVING π, HENG HUAT CHAN π AND π. Itroductio The umber π, as we all ow, is defied to be the legth of a circle of diameter. The first few estimates of π were 3 Egypt aroud 9 B.C.,