# logarithmic

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## 11—20 of 400 matching pages

##### 11: 27.11 Asymptotic Formulas: Partial Sums
27.11.3 $\sum_{n\leq x}\frac{d\left(n\right)}{n}=\frac{1}{2}(\ln x)^{2}+2\gamma\ln x+O% \left(1\right),$
27.11.8 $\sum_{p\leq x}\frac{1}{p}=\ln\ln x+A+O\left(\frac{1}{\ln x}\right),$
27.11.10 $\sum_{p\leq x}\frac{\ln p}{p}=\ln x+O\left(1\right).$
27.11.11 $\sum_{\begin{subarray}{c}p\leq x\\ p\equiv h\!\!\!\!\!\pmod{k}\end{subarray}}\frac{\ln p}{p}=\frac{1}{\phi\left(k% \right)}\ln x+O\left(1\right),$
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 $\left(h,k\right)=1$, then the number of primes $p\leq x$ with $p\equiv h\pmod{k}$ is asymptotic to $x/(\phi\left(k\right)\ln x)$ as $x\to\infty$.
##### 12: 6.15 Sums
6.15.1 $\sum_{n=1}^{\infty}\mathrm{Ci}\left(\pi n\right)=\tfrac{1}{2}(\ln 2-\gamma),$
6.15.2 $\sum_{n=1}^{\infty}\frac{\mathrm{si}\left(\pi n\right)}{n}=\tfrac{1}{2}\pi(\ln% \pi-1),$
6.15.3 $\sum_{n=1}^{\infty}(-1)^{n}\mathrm{Ci}\left(2\pi n\right)=1-\ln 2-\tfrac{1}{2}\gamma,$
6.15.4 $\sum_{n=1}^{\infty}(-1)^{n}\frac{\mathrm{si}\left(2\pi n\right)}{n}=\pi(\tfrac% {3}{2}\ln 2-1).$
##### 15: 4.3 Graphics 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 $-\pi<\Im z<\pi$ onto the whole $w$-plane cut along the negative real axis, where $w=e^{z}$ and $z=\ln w$ (principal value). … Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify 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 $\ln\left(1+z\right)=z-\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3}-\cdots,$ $|z|\leq 1$, $z\neq-1$,
4.6.2 $\ln z=\left(\frac{z-1}{z}\right)+\frac{1}{2}\left(\frac{z-1}{z}\right)^{2}+% \frac{1}{3}\left(\frac{z-1}{z}\right)^{3}+\cdots,$ $\Re z\geq\frac{1}{2}$,
4.6.3 $\ln z=(z-1)-\tfrac{1}{2}(z-1)^{2}+\tfrac{1}{3}(z-1)^{3}-\cdots,$ $|z-1|\leq 1$, $z\neq 0$,
4.6.6 $\ln\left(z+a\right)=\ln a+2\left(\left(\frac{z}{2a+z}\right)+\frac{1}{3}\left(% \frac{z}{2a+z}\right)^{3}+\frac{1}{5}\left(\frac{z}{2a+z}\right)^{5}+\cdots% \right),$ $a>0$, $\Re z\geq-a$, $z\neq-a$.
##### 17: 4.7 Derivatives and Differential Equations
###### §4.7(i) Logarithms
For a nonvanishing analytic function $f(z)$, the general solution of the differential equation …
4.7.6 $w(z)=\operatorname{Ln}\left(f(z)\right)+\hbox{ constant}.$
When $a^{z}$ is a general power, $\ln a$ is replaced by the branch of $\operatorname{Ln}a$ 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 $\mathrm{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …
##### 19: 4.47 Approximations
###### §4.47(i) Chebyshev-Series Expansions
Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. … Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\operatorname{arcsin}$, $\operatorname{arccos}$, $\operatorname{arctan}$, $\sinh$, $\cosh$, $\tanh$, $\operatorname{arcsinh}$, $\operatorname{arccosh}$. … Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\operatorname{arctan}$, $\operatorname{arcsinh}$. …