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6000 DLMF: §6.11 Relations to Other Functions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals

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6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.10 Other Series Expansions 6.12 Asymptotic Expansions

§6.11 Relations to Other Functions

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Notes:: For (6.11.1) and (6.11.2) see Temme (1996b, pp. 180 and 187). For (6.11.3) use (6.5.7).
Permalink:: http://dlmf.nist.gov/6.11
See also:: Annotations for Ch.6

For the notation see §§8.2(i) and 13.2(i).

Incomplete Gamma Function

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Keywords:: exponential integrals, incomplete gamma function, incomplete gamma functions, relations to other functions
See also:: Annotations for §6.11 and Ch.6

6.11.1		$E_{1}\left(z\right)=\Gamma\left(0,z\right).$
ⓘ Symbols: $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function and $z$ : complex variable A&S Ref: 5.1.45 (special case of) Referenced by: §6.11 Permalink: http://dlmf.nist.gov/6.11.E1 Encodings: TeX, pMML, png See also: Annotations for §6.11, §6.11 and Ch.6

Confluent Hypergeometric Function

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Keywords:: auxiliary functions for sine and cosine integrals, confluent hypergeometric functions, exponential integrals, relation to confluent hypergeometric functions, relations to other functions, sine and cosine integrals
See also:: Annotations for §6.11 and Ch.6

6.11.2		$E_{1}\left(z\right)=e^{-z}U\left(1,1,z\right),$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral and $z$ : complex variable Referenced by: §6.11, 5th item, §6.6 Permalink: http://dlmf.nist.gov/6.11.E2 Encodings: TeX, pMML, png See also: Annotations for §6.11, §6.11 and Ch.6

6.11.3		$\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=U\left(1,1,-iz\right).$
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\mathrm{i}$ : imaginary unit, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.11, 5th item Permalink: http://dlmf.nist.gov/6.11.E3 Encodings: TeX, pMML, png See also: Annotations for §6.11, §6.11 and Ch.6

6.10 Other Series Expansions 6.12 Asymptotic Expansions

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