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1: 4.8 Identities
4.8.1 Ln ( z 1 z 2 ) = Ln z 1 + Ln z 2 .
This is interpreted that every value of Ln ( z 1 z 2 ) is one of the values of Ln z 1 + Ln z 2 , and vice versa. …
4.8.3 Ln z 1 z 2 = Ln z 1 Ln z 2 ,
4.8.10 exp ( ln z ) = exp ( Ln z ) = z .
where the integer k is chosen so that ( i z ln a ) + 2 k π [ π , π ] . …
2: 4.10 Integrals
4.10.1 d z z = ln z ,
4.10.2 ln z d z = z ln z z ,
4.10.3 z n ln z d z = z n + 1 n + 1 ln z z n + 1 ( n + 1 ) 2 , n 1 ,
4.10.4 d z z ln z = ln ( ln z ) ,
4.10.13 0 d x e x + 1 = ln 2 .
3: 4.12 Generalized Logarithms and Exponentials
4.12.1 ϕ ( x + 1 ) = e ϕ ( x ) , 1 < x < ,
4.12.6 ϕ ( x ) = ln ( x + 1 ) , 1 < x < 0 ,
4.12.8 ψ ( x ) = e x 1 , < x < 0 ,
4.12.9 ψ ( x ) = + ln ln  times x , x > 1 ,
4.12.10 0 ln ln times x < 1 .
4: 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 + ,
5: 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 ,
4.5.4 ln x x 1 , x > 0 ,
4.5.5 ln x a ( x 1 / a 1 ) , a , x > 0 ,
4.5.6 | ln ( 1 + z ) | ln ( 1 | z | ) , | z | < 1 .
6: 4.2 Definitions
The general logarithm function Ln z is defined by … The real and imaginary parts of ln z are given by … The only zero of ln z is at z = 1 . … Consequently ln z is two-valued on the cut, and discontinuous across the cut. … Natural logarithms have as base the unique positive number …
7: 6.8 Inequalities
6.8.1 1 2 ln ( 1 + 2 x ) < e x E 1 ( x ) < ln ( 1 + 1 x ) ,
6.8.2 x x + 1 < x e x E 1 ( x ) < x + 1 x + 2 ,
6.8.3 x ( x + 3 ) x 2 + 4 x + 2 < x e x E 1 ( x ) < x 2 + 5 x + 2 x 2 + 6 x + 6 .
8: 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 ) .
9: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and e x . … Magnify
Figure 4.3.2 illustrates the conformal mapping of the strip π < z < π onto the whole w -plane cut along the negative real axis, where w = e z and z = ln w (principal value). …
See accompanying text
Figure 4.3.2: Conformal mapping of exponential and logarithm. w = e z , z = ln w . Magnify
See accompanying text
Figure 4.3.3: ln ( x + i y ) (principal value). … Magnify 3D Help
10: 5.17 Barnes’ G -Function (Double Gamma Function)
5.17.4 Ln G ( z + 1 ) = 1 2 z ln ( 2 π ) 1 2 z ( z + 1 ) + z Ln Γ ( z + 1 ) 0 z Ln Γ ( t + 1 ) d t .
In this equation (and in (5.17.5) below), the Ln ’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i). …
5.17.5 Ln G ( z + 1 ) 1 4 z 2 + z Ln Γ ( z + 1 ) ( 1 2 z ( z + 1 ) + 1 12 ) ln z ln A + k = 1 B 2 k + 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) z 2 k .
5.17.6 A = e C = 1.28242 71291 00622 63687 ,
5.17.7 C = lim n ( k = 1 n k ln k ( 1 2 n 2 + 1 2 n + 1 12 ) ln n + 1 4 n 2 ) = γ + ln ( 2 π ) 12 ζ ( 2 ) 2 π 2 = 1 12 ζ ( 1 ) ,