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1: 6.15 Sums
6.15.2 n = 1 si ( π n ) n = 1 2 π ( ln π 1 ) ,
6.15.3 n = 1 ( 1 ) n Ci ( 2 π n ) = 1 ln 2 1 2 γ ,
6.15.4 n = 1 ( 1 ) n si ( 2 π n ) n = π ( 3 2 ln 2 1 ) .
2: 4.10 Integrals
4.10.1 d z z = ln z ,
4.10.2 ln z d z = z ln z z ,
4.10.4 d z z ln z = ln ( ln z ) ,
4.10.5 0 1 ln t 1 t d t = π 2 6 ,
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
3: 4.1 Special Notation
k , m , n

integers.

4: 4.6 Power Series
4.6.1 ln ( 1 + z ) = z 1 2 z 2 + 1 3 z 3 , | z | 1 , z 1 ,
4.6.2 ln z = ( z 1 z ) + 1 2 ( z 1 z ) 2 + 1 3 ( z 1 z ) 3 + , z 1 2 ,
4.6.3 ln z = ( z 1 ) 1 2 ( z 1 ) 2 + 1 3 ( z 1 ) 3 , | z 1 | 1 , z 0 ,
4.6.4 ln z = 2 ( ( z 1 z + 1 ) + 1 3 ( z 1 z + 1 ) 3 + 1 5 ( z 1 z + 1 ) 5 + ) , z 0 , z 0 ,
4.6.5 ln ( z + 1 z 1 ) = 2 ( 1 z + 1 3 z 3 + 1 5 z 5 + ) , | z | 1 , z ± 1 ,
5: 6.6 Power Series
6.6.1 Ei ( x ) = γ + ln x + n = 1 x n n ! n , x > 0 .
6.6.2 E 1 ( z ) = γ ln z n = 1 ( 1 ) n z n n ! n .
6.6.6 Ci ( z ) = γ + ln z + n = 1 ( 1 ) n z 2 n ( 2 n ) ! ( 2 n ) .
6: 4.5 Inequalities
4.5.1 x 1 + x < ln ( 1 + x ) < x , x > 1 , x 0 ,
4.5.2 x < ln ( 1 x ) < x 1 x , x < 1 , x 0 ,
4.5.3 | ln ( 1 x ) | < 3 2 x , 0 < x 0.5828 ,
4.5.4 ln x x 1 , x > 0 ,
4.5.5 ln x a ( x 1 / a 1 ) , a , x > 0 ,
7: 4.2 Definitions
This is a multivalued function of z with branch point at z = 0 . The principal value, or principal branch, is defined by
4.2.2 ln z = 1 z d t t ,
4.2.12 ln e = 1 .
4.2.16 ln z = ( ln 10 ) log 10 z ,
8: 22.14 Integrals
22.14.1 sn ( x , k ) d x = k 1 ln ( dn ( x , k ) k cn ( x , k ) ) ,
22.14.4 cd ( x , k ) d x = k 1 ln ( nd ( x , k ) + k sd ( x , k ) ) ,
9: 4.8 Identities
4.8.7 ln 1 z = ln z , | ph z | π .
4.8.10 exp ( ln z ) = exp ( Ln z ) = z .
4.8.13 ln ( a x ) = x ln a , a > 0 .
10: 5.10 Continued Fractions
5.10.1 Ln Γ ( z ) + z ( z 1 2 ) ln z 1 2 ln ( 2 π ) = a 0 z + a 1 z + a 2 z + a 3 z + a 4 z + a 5 z + ,