About the Project
NIST

logarithmic

AdvancedHelp

(0.001 seconds)

11—20 of 400 matching pages

11: 27.11 Asymptotic Formulas: Partial Sums
27.11.3 n x d ( n ) n = 1 2 ( ln x ) 2 + 2 γ ln x + O ( 1 ) ,
27.11.8 p x 1 p = ln ln x + A + O ( 1 ln x ) ,
27.11.10 p x ln p p = ln x + O ( 1 ) .
27.11.11 p x p h ( mod k ) ln p p = 1 ϕ ( k ) ln x + O ( 1 ) ,
The prime number theorem for arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if ( h , k ) = 1 , then the number of primes p x with p h ( mod k ) is asymptotic to x / ( ϕ ( k ) ln x ) as x .
12: 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 ) .
13: 27 Functions of Number Theory
14: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals
Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals
15: 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
16: 4.6 Power Series
§4.6(i) Logarithms
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 ,
4.6.6 ln ( z + a ) = ln a + 2 ( ( z 2 a + z ) + 1 3 ( z 2 a + z ) 3 + 1 5 ( z 2 a + z ) 5 + ) , a > 0 , z - a , z - a .
17: 4.7 Derivatives and Differential Equations
§4.7(i) Logarithms
4.7.1 d d z ln z = 1 z ,
For a nonvanishing analytic function f ( z ) , the general solution of the differential equation …
4.7.6 w ( z ) = Ln ( f ( z ) ) +  constant .
When a z is a general power, ln a is replaced by the branch of Ln a used in constructing a z . …
18: 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 ) . …
19: 4.47 Approximations
§4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for ln , exp , sin , cos , tan , cot , arcsin , arctan , arcsinh . … Hart et al. (1968) give ln , exp , sin , cos , tan , cot , arcsin , arccos , arctan , sinh , cosh , tanh , arcsinh , arccosh . … Luke (1975, Chapter 3) supplies real and complex approximations for ln , exp , sin , cos , tan , arctan , arcsinh . …
20: 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 .