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1: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
As in the case of the logarithm4.2(i)) there is a cut along the interval ( , 0 ] and the principal value is two-valued on ( , 0 ) . … The logarithmic integral is defined by
6.2.8 li ( x ) = 0 x d t ln t = Ei ( ln x ) , x > 1 .
2: 4.11 Sums
§4.11 Sums
For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).
3: 4.12 Generalized Logarithms and Exponentials
§4.12 Generalized Logarithms and Exponentials
A generalized exponential function ϕ ( x ) satisfies the equations …Its inverse ψ ( x ) is called a generalized logarithm. It, too, is strictly increasing when 0 x 1 , and … For analytic generalized logarithms, see Kneser (1950).
4: 4.2 Definitions
§4.2(i) The Logarithm
The general logarithm function Ln z is defined by …
§4.2(ii) Logarithms to a General Base a
Natural logarithms have as base the unique positive number …
5: 4.10 Integrals
§4.10(i) Logarithms
4.10.4 d z z ln z = ln ( ln z ) ,
4.10.7 0 x d t ln t = li ( x ) , x > 1 .
For li ( x ) see §6.2(i). … Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).
6: 4.8 Identities
§4.8(i) Logarithms
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. …
4.8.10 exp ( ln z ) = exp ( Ln z ) = z .
where the integer k is chosen so that ( i z ln a ) + 2 k π [ π , π ] .
4.8.13 ln ( a x ) = x ln a , a > 0 .
7: 5.10 Continued Fractions
§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 + ,
8: 27.12 Asymptotic Formulas: Primes
27.12.1 lim n p n n ln n = 1 ,
27.12.2 p n > n ln n , n = 1 , 2 , .
For the logarithmic integral li ( x ) see (6.2.8). … π ( x ) li ( x ) changes sign infinitely often as x ; see Littlewood (1914), Bays and Hudson (2000). …
27.12.7 | π ( x ) li ( x ) | < 1 8 π x ln x .
9: 4.44 Other Applications
§4.44 Other Applications
The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. … For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …
10: 4.3 Graphics
See accompanying text
Figure 4.3.1: ln x and e x . … Magnify
§4.3(ii) Complex Arguments: Conformal Maps
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. … Magnify
See accompanying text
Figure 4.3.3: ln ( x + i y ) (principal value). … Magnify 3D Help