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1: 6.2 Definitions and Interrelations
§6.2(i) Exponential and Logarithmic Integrals
The logarithmic integral is defined by
6.2.8 li ( x ) = 0 x d t ln t = Ei ( ln x ) , x > 1 .
6.2.13 Ci ( z ) = - Cin ( z ) + ln z + γ .
6.2.16 Chi ( z ) = γ + ln z + 0 z cosh t - 1 t d t .
2: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals Ei ( x ) , E 1 ( z ) , and Ein ( z ) ; the logarithmic integral li ( x ) ; the sine integrals Si ( z ) and si ( z ) ; the cosine integrals Ci ( z ) and Cin ( z ) . …
3: 6.16 Mathematical Applications
§6.16(ii) Number-Theoretic Significance of li ( x )
If we assume Riemann’s hypothesis that all nonreal zeros of ζ ( s ) have real part of 1 2 25.10(i)), then
6.16.5 li ( x ) - π ( x ) = O ( x ln x ) , x ,
See accompanying text
Figure 6.16.2: The logarithmic integral li ( x ) , together with vertical bars indicating the value of π ( x ) for x = 10 , 20 , , 1000 . Magnify
4: 27.12 Asymptotic Formulas: Primes
27.12.5 | π ( x ) - li ( x ) | = O ( x exp ( - c ( ln x ) 1 / 2 ) ) , x .
For the logarithmic integral li ( x ) see (6.2.8). …
27.12.6 | π ( x ) - li ( x ) | = O ( x exp ( - d ( ln x ) 3 / 5 ( ln ln x ) - 1 / 5 ) ) .
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 .
5: 4.10 Integrals
§4.10(i) Logarithms
4.10.2 ln z d z = z ln z - 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: 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 .
7: 27.21 Tables
Bressoud and Wagon (2000, pp. 103–104) supplies tables and graphs that compare π ( x ) , x / ln x , and li ( x ) . …
8: 6.3 Graphics
For a graph of li ( x ) see Figure 6.16.2. …
See accompanying text
Figure 6.3.3: | E 1 ( x + i y ) | , - 4 x 4 , - 4 y 4 . …Also, | E 1 ( z ) | logarithmically as z 0 . Magnify 3D Help
9: 6.14 Integrals
6.14.1 0 e - a t E 1 ( t ) d t = 1 a ln ( 1 + a ) , a > - 1 ,
6.14.2 0 e - a t Ci ( t ) d t = - 1 2 a ln ( 1 + a 2 ) , a > 0 ,
6.14.3 0 e - a t si ( t ) d t = - 1 a arctan a , a > 0 .
6.14.4 0 E 1 2 ( t ) d t = 2 ln 2 ,
6.14.7 0 Ci ( t ) si ( t ) d t = ln 2 .
10: 6.12 Asymptotic Expansions
§6.12(i) Exponential and Logarithmic Integrals
6.12.2 Ei ( x ) e x x ( 1 + 1 ! x + 2 ! x 2 + 3 ! x 3 + ) , x + .
For the function χ see §9.7(i). The asymptotic expansion of li ( x ) as x is obtainable from (6.2.8) and (6.12.2). …
6.12.7 R n ( f ) ( z ) = ( - 1 ) n 0 e - z t t 2 n t 2 + 1 d t ,