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
 =
 óõ 1 0 dt 1 + t
 log 2
 =
 9 16 + 2 óõ 3/4 0 æè Ö 1 + t2 -1 öø dt
 log 2
 =
 15 16 - 8 óõ 1/8 0 Ö t + t2 dt
 log 2
 =
 óõ 1 0 t2n + 1 - tn log t dt       n ³ 0
 log 2
 =
 1 + óõ p/2 0 sin t log sin t dt
 log 2
 =
 óõ ¥ 0 cos t - cos(2t) t dt
 log 2
 =
 1 2 óõ ¥ 0 t cosh t + 1 dt
 p 2 log 2
 =
 óõ 1 0 arcsin t t dt
 p 2 log 2
 =
 óõ ¥ 0 arctan(2t) - arctan t t dt
 - p 2 log 2
 =
 óõ p/2 0 log sin t dt
 p 8 log 2
 =
 óõ p/4 0 log(1 + tan t)dt
 2-2log 2- p2 12
 =
 óõ 1 0 log t log(1 + t) dt
 - log22 2
 =
 óõ 1/2 0 log(1 - t) 1 - t dt
 log2 2 2 - p2 12
 =
 óõ 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 æè 1- 1 12 + 1.2 12.20 - 1.2.3 12.20.28 +... öø
 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 æè 1+ æè 1+ 1 3 öø 1 9 + æè 1+ 1 3 + 1 5 öø 1 92 +... öø
 log 2
 =
 2 3 + ¥å k=1 æè 1 2k + 1 4k+1 + 1 8k+4 + 1 16k+12 öø 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 æè 10 9 öø -2log æè 25 24 öø +3log æè 81 80 öø (Adams [1])
 log 2
 =
 1 2 æè 1 2 + 1.3 2.4 1 2 + 1.3.5 2.4.6 1 3 +... öø (Nielsen [4])
 log 2
 =
 lim k® ¥ æè 1 k+1 + 1 k+2 +...+ 1 2k öø

2.2  Machin like series

In this part we use the following notation
 L(p)=arctanh æè 1 p öø = 1 2 log æè p + 1 p - 1 öø .

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= Nå 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).

Back to Numbers, Constants and Computation
File translated from TEX by TTH, version 3.01.
On 6 Sep 2001, 22:33.