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 $\frac{1}{2}$ (§25.10(i)), then …

3: 6.11 Relations to Other Functions

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6.11.2 $$E_1\left(z\right)={\mathrm{e}}^{-z}U(1,1,z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral and $z$: complex variable

Referenced by:

§6.11, 5th item, §6.6

Permalink:

http://dlmf.nist.gov/6.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.11, §6.11 and Ch.6

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4: 25.12 Polylogarithms

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. ►

Integral Representation

… ►Further properties include …and … ►In terms of polylogarithms …

5: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

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§8.19(v) Recurrence Relation and Derivatives

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§8.19(vi) Relation to Confluent Hypergeometric Function

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§8.19(x) Integrals

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6: 7.5 Interrelations

§7.5 Interrelations

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7.5.13 $$G\left(x\right)=\sqrt{\pi }F\left(x\right)-\frac{1}{2}{\mathrm{e}}^{-x^2}\mathrm{Ei}\left(x^2\right),$$ $x>0$.

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Symbols:

$F\left(z\right)$: Dawson’s integral, $G\left(z\right)$: Goodwin–Staton integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral and $x$: real variable

Referenced by:

§7.5

Permalink:

http://dlmf.nist.gov/7.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §7.5 and Ch.7

►For $\mathrm{Ei}\left(x\right)$ see §6.2(i).

7: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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8: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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9: 19.10 Relations to Other Functions

§19.10 Relations to Other Functions

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§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)). …