# relation to logarithmic integral

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##### 1: 6.2 Definitions and Interrelations
The logarithmic integral is defined by …
##### 2: 6.16 Mathematical Applications
If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta\left(s\right)$ have real part of $\tfrac{1}{2}$25.10(i)), then …
##### 4: 25.12 Polylogarithms
The special case $z=1$ is the Riemann zeta function: $\zeta\left(s\right)=\operatorname{Li}_{s}\left(1\right)$.
###### Integral Representation
Further properties include …and … In terms of polylogarithms …
##### 6: 7.5 Interrelations
###### §7.5 Interrelations
7.5.13 $G\left(x\right)=\sqrt{\pi}F\left(x\right)-\tfrac{1}{2}e^{-x^{2}}\operatorname{% Ei}\left(x^{2}\right),$ $x>0$.
For $\operatorname{Ei}\left(x\right)$ see §6.2(i).
##### 9: 19.10 Relations to Other Functions
###### §19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
###### §19.10(ii) Elementary Functions
In each case when $y=1$, the quantity multiplying $R_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …
##### 10: 15.17 Mathematical Applications
This topic is treated in §§15.10 and 15.11. The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. …
###### §15.17(ii) Conformal Mappings
Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …