Collection of formulae for log 2

1  Integral formulae

The constant log 2 occurs in the exact value of many integrals, we give a selection of such integrals.

log 2 = ó õ 1 0  dt 1 + t log 2 = 9 16  + 2 ó õ 3/4 0  æ è Ö   1 + t2   -1 ö ø dt log 2 = 15 16  - 8 ó õ 1/8 0  Ö   t + t2   dt log 2 = ó õ 1 0  t2n + 1 - tn log t dt       n ³ 0 log 2 = 1 +    ó õ p/2 0  sin t log sin t dt log 2 = ó õ ¥ 0  cos t - cos(2t) t dt log 2 =  1 2 ó õ ¥ 0   t cosh t + 1 dt p 2 log 2 = ó õ 1 0  arcsin t t dt p 2 log 2 = ó õ ¥ 0  arctan(2t) - arctan t t dt - p 2 log 2 = ó õ p/2 0  log sin t dt p 8 log 2 = ó õ p/4 0  log(1 + tan t)dt 2-2log 2- p2 12 = ó õ 1 0  log t log(1 + t) dt - log22 2 = ó õ 1/2 0  log(1 - t) 1 - t dt log2 2 2 - p2 12 = ó õ 1/2 0  log(1 - t) t dt

2  Series formulae

Now here is a list of series for log 2.

2.1  Series

log 2 = 1- 1 2 + 1 3 - 1 4 +...       (Mercator) log 2 = 1 2 ¥ å k=1  1 k(2k-1)        (Mercator) log 2 = 3 4 - 1 4 ¥ å k=1  1 k(k+1)(2k+1) log 2 = 1 2 + ¥ å k=0   1 (2k+1)(2k+2)(2k+3)        (Knopp [3]) log 2 = 1 2 + 1 2 ¥ å k=1  1 k(4k2-1)        (Ramanujan) log 2 = 2 3 + 1 3 ¥ å k=1   1 k(16k2-1)        (Ramanujan) log 2 = 3 4 - 1 2 ¥ å k=1  1 k(4k2-1)2        (Knopp [3]) log 2 = 1 2 + ¥ å k=1  (-1)k-1 k(4k4+1)        (Glaisher) log 2 = 1327 1920 + 45 4 ¥ å k=4   (-1)k k(k2-1)(k2-4)(k2-9)        (Glaisher [2]) log 2 = ¥ å k=1  z(2k)-1 k log 2 = 1 2 + ¥ å k=2  z(k)-1 2k log 2 = 31 45 + 1 3 ¥ å k=1  z(2k+1)-1 16k log 2 = 3 4 æ è 1- 1 12 + 1.2 12.20 - 1.2.3 12.20.28 +... ö ø log 2 = ¥ å k=1  1 k2k log 2 = 1 2 ¥ å k=1  Hk 2k ,    Hk=1+ 1 2 +...+ 1 k log 2 = 2 3 ¥ å k=0  1 (2k+1)9k log 2 = 16 27 æ è 1+ æ è 1+ 1 3 ö ø 1 9 + æ è 1+ 1 3 + 1 5 ö ø 1 92 +... ö ø log 2 = 2 3 + ¥ å k=1  æ è 1 2k + 1 4k+1 + 1 8k+4 + 1 16k+12 ö ø 1 16k log 2 = 3 4 + 1 4 ¥ å k=1  (-1)k(5k+1)(2k)! k(2k+1)16k(k!)2        (Knopp [3]) log 2 = 7log æ è 10 9 ö ø -2log æ è 25 24 ö ø +3log æ è 81 80 ö ø        (Adams [1]) log 2 = 1 2 æ è 1 2 + 1.3 2.4 1 2 + 1.3.5 2.4.6 1 3 +... ö ø        (Nielsen [4]) log 2 = lim k® ¥  æ è 1 k+1 + 1 k+2 +...+ 1 2k ö ø

2.2  Machin like series

In this part we use the following notation

L(p)=arctanh æ è 1 p ö ø = 1 2 log æ è p + 1 p - 1 ö ø .

This is just a small selection of fast converging formulae (most are from [5]). To give an idea of the efficiency the Lehmer's measure E is also computed for each formula,

E= N å i=1   1 log10(pi2)

where log₁₀ is the common logarithm. The more E is small the more the formula is efficient (this is of course a rough estimate).

2.2.1  Two terms (N=2)

log 2 = 2L(5)+2L(7)    E=1.307 log 2 = 4L(29/5)-2L(577)    E=0.836 log 2 = 4L(6)+2L(99)    E=0.893 log 2 = 4L(7)+2L(17)    E=0.998 log 2 = 6L(9)+2L(253/3)    E=0.784 log 2 = 10L(17)+4L(499/13)    E=0.722

2.2.2  Three terms (N=3)

log 2 = 8L(11)-4L(111)-2L(19601)    E=0.841 log 2 = 10L(17)+8L(79)+4L(1351)    E=0.830 log 2 = 14L(23)+6L(65)-4L(485)    E=0.829 log 2 = 18L(26)-2L(4801)+8L(8749)    E=0.616 log 2 = 14L(31)+10L(49)+6L(161)    E=0.858

2.2.3  Four terms and more (N ³ 4)

log 2 = 36L(52)-2L(4801)+8L(8749)+18L(70226)    E=0.657 log 2 = 72L(127)+54L(449)+34L(4801)-10L(8749)     E=0.689 log 2 = 144L(251)+54L(449)-38L(4801)+62L(8749)    E=0.659 log 2 = 342L(575)+198L(3361)+222L(8749)+160L(13121)+106L(56251)   E=0.676

References

[1]

J.C. Adams, On the value of Euler's constant, Proc. Roy. Soc. London, (1878), vol. 27, p. 88-94

[2]

J.W.L. Glaisher, Methods of increasing the convergence of certain series of reciprocals, Quart. J., (1902), vol. 34, p. 252-347

[3]

K. Knopp, Theory and application of infinite series, Blackie & Son, London, (1951)

[4]

N. Nielsen, Om log(2) og 1/1²-1/3²+1/5²-1/7²+...,Nyt Tidss. for Math., (1894), p. 22-25

[5]

P. Sebah, Machin like formulae for logarithm, Unpublished, (1997).