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principal branches (or values)

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1: 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 ,
2: 4.13 Lambert W -Function
We call the solution for which W ( x ) W ( - 1 / e ) the principal branch and denote it by Wp ( x ) . …
See accompanying text
Figure 4.13.1: Branches Wp ( x ) and Wm ( x ) of the Lambert W -function. … Magnify
4.13.5 Wp ( x ) = n = 1 ( - 1 ) n - 1 n n - 2 ( n - 1 ) ! x n , | x | < 1 e .
4.13.10 Wp ( x ) = ξ - ln ξ + ln ξ ξ + ( ln ξ ) 2 2 ξ 2 - ln ξ ξ 2 + O ( ( ln ξ ) 3 ξ 3 ) ,
4.13.11 Wm ( x ) = - η - ln η - ln η η - ( ln η ) 2 2 η 2 - ln η η 2 + O ( ( ln η ) 3 η 3 ) ,
3: 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 ) .
4: 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 ) ,
4.10.6 0 1 ln t 1 + t d t = - π 2 12 ,
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
5: 4.1 Special Notation
k , m , n integers.
6: 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 ,
7: 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 ) .
8: 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 ,
9: 4.4 Special Values and Limits
4.4.1 ln 1 = 0 ,
4.4.2 ln ( - 1 ± i 0 ) = ± π i ,
4.4.3 ln ( ± i ) = ± 1 2 π i .
4.4.13 lim x x - a ln x = 0 , a > 0 ,
4.4.14 lim x 0 x a ln x = 0 , a > 0 ,
10: 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 ) ) ,