A catalogue of mathematical formulas involving π, with analysis David H. Bailey∗ April 22, 2020 Abstract This paper presents a catalogue of mathematical formulas and iterative algorithms for evaluating the math- ematical constant π, ranging from Archimedes' 2200-year-old iteration to some formulas that were discovered only in the past few decades. Computer implementations and timing results for these formulas and algorithms are also included. In particular, timings are presented for evaluations of various infinite series formulas to approximately 10,000-digit precision, for evaluations of various integral formulas to approximately 4,000-digit precision, and for evaluations of several iterative algorithms to approximately 100,000-digit precision, all based on carefully designed comparative computer runs. 1 Background The mathematical constant known as π = 3:141592653589793 ::: is undeniably the most famous and arguably the most important mathematical constant. Mathematicians since the days of Archimedes, up to and including the present day, have analyzed its properties and computed its numerical value. This paper presents a collection of 72 formulas and algorithms that have been found by mathematicians over the years involving π. While a comprehensive collection is of course not possible, preference is given in this collection for formulas that satisfy the following criteria: • Formulas that give π or a very simple expression involving π explicitly, as opposed to implicit relations such as eiπ + 1 = 0. • Formulas that give π or a very simple expression involving π as an infinite series, definite integral or simple iterative algorithm. • Formulas that involve simple notation, such as summations, integrals, binomial coefficients, exponentials, logarithms, etc., that would be familiar to anyone who has completed a beginning calculus course. • Formulas that are relatively new, discovered within the last 100 years or so. Included in this listing are several formulas for π that have actually have been used in large calculations of π, both before and since the rise of computer technology. These include formulas 2 through 5 prior to the 20th century, and formulas 6, 7, 11, 12, 13, 14, 16, 18, 69 and 71 in the late 20th and early 21st century. Formulas 13 through 18 have the intriguing property that they permit digits (in certain specific bases) of the constant specified on the left-hand side to be calculated beginning at an arbitrary starting position, without needing to calculate any of the digits that came before, by means of relatively simple algorithms. Formulas 13 and 14 have been used in computations of high-order binary digits of π [15, Sec 3.4{3.6], while ∗Lawrence Berkeley National Laboratory, Berkeley, CA 94720 (retired) and University of California, Davis, Department of Com- puter Science, [email protected]. 1 formula 16 has been used in computations of high-order binary digits of π2, and formula 18 has been used in computations of high-order base-3 digits of π2 [11]. Numerous other recently-discovered formulas that possess the arbitrary digit-computation property for various mathematical constants are catalogued in [3]. Many of these formulas are relatively new, in the sense that they were discovered only in the past few decades. The formulas mentioned in the previous paragraph are certainly in this category, having been discovered only since 1996. Many of the formulas from 19 through 50 were not well known until recently. Formulas 64 through 67 are also relatively new, in the sense that they are part of a class of integral formulas that are the subject of current research [8, 9, 10]. Formula 69 was discovered in 1976. Formulas 70, 71 and 72 were first published in 1984. 2 A catalogue of formulas for π 1 π X (−1)n = (1) 4 (2n + 1) n=0 1 1 π X (−1)n X (−1)n = + (2) 4 (2n + 1)22n+1 (2n + 1)32n+1 n=0 n=0 1 1 π X (−1)n X (−1)n = 4 − (3) 4 (2n + 1)52n+1 (2n + 1)2392n+1 n=0 n=0 1 1 1 π X (−1)n X (−1)n X (−1)n = + + (4) 4 (2n + 1)22n+1 (2n + 1)52n+1 (2n + 1)82n+1 n=0 n=0 n=0 1 1 1 π X (−1)n X (−1)n X (−1)n = 3 + + (5) 4 (2n + 1)42n+1 (2n + 1)202n+1 (2n + 1)19852n+1 n=0 n=0 n=0 1 1 1 1 π X (−1)n X (−1)n X (−1)n X (−1)n = 12 + 32 − 5 + 12 4 (2n + 1)492n+1 (2n + 1)572n+1 (2n + 1)2392n+1 (2n + 1)1104432n+1 n=0 n=0 n=0 n=0 (6) 1 1 1 1 π X (−1)n X (−1)n X (−1)n X (−1)n = 44 + 7 − 12 + 24 4 (2n + 1)572n+1 (2n + 1)2392n+1 (2n + 1)6822n+1 (2n + 1)129432n+1 n=0 n=0 n=0 n=0 (7) 1 p X (−1)n π = 12 (8) (2n + 1)3n n=0 p 1 2n 3 3 X π = − 24 n (9) 4 (2n + 3)(2n − 1)42n+1 n=0 1 π Y 4n2 = (10) 2 4n2 − 1 n=1 p 1 1 2 2 X (4n)!(1103 + 26390n) = (11) π 9801 (n!)43964n n=0 1 1 X (−1)n(6n)!(13591409 + 545140134n) = 12 (12) π (3n)!(n!)36403203n+3=2 n=0 2 1 X 1 4 2 1 1 π = − − − (13) 16n 8n + 1 8n + 4 8n + 5 8n + 6 n=0 1 1 X (−1)n 1 X (−1)n 32 8 1 π = 4 − + + (14) 4n(2n + 1) 64 1024n 4n + 1 4n + 2 4n + 3 n=0 n=0 1 X (−1)n 2 2 1 π = + + (15) 4n 4n + 1 4n + 2 4n + 3 n=0 1 2 9 X 1 16 24 8 6 1 π = − − − + (16) 8 64n (6n + 1)2 (6n + 2)2 (6n + 3)2 (6n + 4)2 (6n + 5)2 n=0 1 p 1 X 1 81 54 9 12 3 2 1 π 3 = − − − − − − 9 729n 12n + 1 12n + 2 12n + 4 12n + 6 12n + 7 12n + 8 12n + 10 n=0 (17) 1 2 2 X 1 243 405 81 27 72 π = − − − − 27 729n (12n + 1)2 (12n + 2)2 (12n + 4)2 (12n + 5)2 (12n + 6)2 n=0 9 9 5 1 − − − + (18) (12n + 7)2 (12n + 8)2 (12n + 10)2 (12n + 11)2 1 X 12n22n 3π + 8 = 4n (19) n=0 2n 2 1 2n π 2 X − 2 log 2 = n (20) 6 n24n n=1 1 X (126n2 − 24n + 8)23n 15π + 52 = 6n (21) n=0 3n 1 X (1920n3 − 928n2 + 424n − 16)24n 105π + 304 = 8n (22) n=0 4n 1 p X (49n + 1)8n 16π 3 + 81 = (23) n3n n=0 3 n 1 p X (−245n + 338)8n 162 − 6π 3 − 18 log 3 = (24) n3n n=0 3 n 1 X (50n − 6) π = (25) n3n n=0 2 n 1 X (−4)n(2n)!2(3n)!(201 − 952n) 15π + 42 = (26) (6n)!n! n=1 1 p X 8n(2n)!2(3n)!(350n − 17) 15π 2 + 27 = (27) (6n)!n! n=0 1 p X (−27)n(2n)!2(3n)!(81 − 1080n) 40π 3 + 243 = (28) (6n)!n! n=1 1 p X (−1=3)n(2n)!2(3n)!(4123 − 22100n) 20π 3 + 89 = (29) (6n)!n! n=1 3 1 X (−1=2)n(89012n3 − 77362n2 + 482n + 3028) 15π + 240 log 2 − 528 = 5n (30) n=1 2n 1 p X 9n(2743n2 − 130971n − 12724) 24516 − 360π 3 = 4n (31) n=1 n 1 X 8n(430n2 − 6240n − 520) 45π + 1164 = 4n (32) n=1 n 1 p X 3n(7175n2 − 15215n + 480) 40π 3 + 1872 = 4n (33) n=1 n 1 p X (9=8)n(5692 + 6335n − 5415n2) 288π 3 − 576 log 2 + 324 = 4n (34) n=0 n 1 p X (9=8)n(7517 + 1145n + 18050n2) 1008π 3 − 576 log 2 + 7587 = 4n (35) n=0 n 1 !3 16 X 42n + 5 2n = (36) π 4096n n n=0 1 4 X (−1)n(4n)!(20n + 3) = (37) π 44n(n!)422n+1 n=0 1 4 X (−1)n(4n)!(260n + 23) = (38) π 44n(n!)4182n+1 n=0 1 4 X (−1)n(4n)!(21460n + 1123) = (39) π 44n(n!)48822n+1 n=0 1 2 X (4n)!(8n + 1) p = (40) 44n(n!)432n+1 π 3 n=0 1 1 X (4n)!(10n + 1) p = (41) 44n(n!)492n+1 2π 2 n=0 1 4 X (−1)n(4n)!(28n + 3) p = (42) 44n(n!)43n42n+1 π 3 n=0 1 4 X (−1)n(4n)!(644n + 41) p = (43) 44n(n!)45n722n+1 π 5 n=0 1 1 X (4n)!(40n + 3) p = (44) 44n(n!)4492n+1 3π 3 n=0 1 4n2n4 2 32 X (120n + 34n + 3) = 2n n (45) π2 216n n=0 1 n2n5 2 128 X (−1) (820n + 180n + 13) = n (46) π2 220n n=0 1 !3 2 X n 2n 4n + 1 = (−1) (47) π n 64n n=0 4 1 !3 4 X 2n 6n + 1 = (48) π n 256n n=0 1 X 2n+1 π + 4 = 2n (49) n=0 n 1 6 X (6n)!(532n2 + 126n + 9) = 64 (50) π2 (n!)6106n+3 n=0 π Z 1 dx = 2 (51) 4 0 1 + x π Z 1 p = 1 − x2 dx (52) 4 0 π − 2 Z 1 = x tan−1 x dx (53) 4 0 π(π − 12) log 2 Z 1 + = log x tan−1 x dx (54) 48 2 0 22 Z 1 x4(1 − x)4 dx − π = 2 (55) 7 0 1 + x π Z 1 x2 dx = p (56) 4 4 8 0 (1 + x ) 1 − x π(1 + 2 log 2) Z 1 p = xe−x 1 − e−2x dx (57) 8 0 π3 Z 1 x2 dx 4π log2 2 + = p (58) x 3 0 e − 1 Z π=2 x2 dx π log 2 = 2 (59) 0 sin x π3 π log2 2 Z π=2 + = log2(cos x) dx (60) 24 2 0 8π3 Z 1 log2 x dx p = 2 (61) 81 3 0 x + x + 1 π Z 1 log(1 + x2) dx − log 2 = 2 (62) 2 0 x p Z 1 π = Γ(1=2) = x−1=2e−x dx (63) p0 π 3 log(2 + 3) Z 1 Z 1 Z 1 dx dy dz − + = (64) p 2 2 2 4 2 0 0 0 x + y + z p p π 3 log(2 + 3) Z 1 Z 1 Z 1 p − + + = x2 + y2 + z2 dx dy dz (65) 24 4 2 p p 0 0 0 π 2 3 7 log(2 + 3) Z 1 Z 1 Z 1 − + + = (x2 + y2 + z2)3=2dx dy dz (66) 60 5 20 0 0 0 Z 1 Z 1 x − 1 2 y − 1 2 xy − 1 2 5 − π2 − 4 log 2 + 16 log2 2 = dx dy (67) 0 0 x + 1 y + 1 xy + 1 5 3 Iterative algorithms for π p • (The Archimedes iteration).