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
=

1

0 
dt

1 + t
log 2
=
9

16
 + 2
3/4

0 



 

1 + t2
 
-1
dt
log 2
=
15

16
 - 8
1/8

0 


 

t + t2
 
dt
log 2
=

1

0 
t2n + 1 - tn

log t
dt       n 0
log 2
=
1 +   
p/2

0 
sin t log sin t dt
log 2
=



0 
cos t - cos(2t)

t
dt
log 2
=
 1

2



0 
 t

cosh t + 1
dt
p

2
log 2
=

1

0 
arcsin t

t
dt
p

2
log 2
=



0 
arctan(2t) - arctan t

t
dt
- p

2
log 2
=

p/2

0 
log sin t dt
p

8
log 2
=

p/4

0 
log(1 + tan t)dt
2-2log 2- p2

12
=

1

0 
log t log(1 + t) dt
- log22

2
=

1/2

0 
log(1 - t)

1 - t
dt
log2 2

2
- p2

12
=

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


k=1 
1

k(2k-1)
       (Mercator)
log 2
=
3

4
- 1

4


k=1 
1

k(k+1)(2k+1)
log 2
=
1

2
+

k=0 
 1

(2k+1)(2k+2)(2k+3)
       (Knopp [3])
log 2
=
1

2
+ 1

2


k=1 
1

k(4k2-1)
       (Ramanujan)
log 2
=
2

3
+ 1

3


k=1 
 1

k(16k2-1)
       (Ramanujan)

log 2
=
3

4
- 1

2


k=1 
1

k(4k2-1)2
       (Knopp [3])
log 2
=
1

2
+

k=1 
(-1)k-1

k(4k4+1)
       (Glaisher)
log 2
=
1327

1920
+ 45

4


k=4 
 (-1)k

k(k2-1)(k2-4)(k2-9)
       (Glaisher [2])
log 2
=


k=1 
z(2k)-1

k
log 2
=
1

2
+

k=2 
z(k)-1

2k
log 2
=
31

45
+ 1

3


k=1 
z(2k+1)-1

16k
log 2
=
3

4

1- 1

12
+ 1.2

12.20
- 1.2.3

12.20.28
+...
log 2
=


k=1 
1

k2k
log 2
=
1

2


k=1 
Hk

2k
,    Hk=1+ 1

2
+...+ 1

k
log 2
=
2

3


k=0 
1

(2k+1)9k
log 2
=
16

27

1+
1+ 1

3

1

9
+
1+ 1

3
+ 1

5

1

92
+...
log 2
=
2

3
+

k=1 

1

2k
+ 1

4k+1
+ 1

8k+4
+ 1

16k+12

1

16k
log 2
=
3

4
+ 1

4


k=1 
(-1)k(5k+1)(2k)!

k(2k+1)16k(k!)2
       (Knopp [3])
log 2
=
7log
10

9

-2log
25

24

+3log
81

80

       (Adams [1])
log 2
=
1

2

1

2
+ 1.3

2.4
1

2
+ 1.3.5

2.4.6
1

3
+...
       (Nielsen [4])

log 2
=

lim
k  

1

k+1
+ 1

k+2
+...+ 1

2k

2.2  Machin like series

In this part we use the following notation
L(p)=arctanh
1

p

= 1

2
log
p + 1

p - 1

.

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

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 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).


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On 6 Sep 2001, 22:33.