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

§6.11 Relations to Other Functions

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

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

¶ Incomplete Gamma Function

Keywords:: exponential integrals, incomplete gamma functions

6.11.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}\nolimits\!\left(0,% z\right).$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function and : complex variable
A&S Ref:: 5.1.45 (special case of)
Referenced by:: §6.11
Permalink:: http://dlmf.nist.gov/6.11.E1
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¶ Confluent Hypergeometric Function

Keywords:: auxiliary functions for sine and cosine integrals, confluent hypergeometric functions, exponential integrals

6.11.2 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=e^{{-z}}\mathop{U\/}\nolimits\!\left% (1,1,z\right),$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral and : complex variable
Referenced by:: §6.11, §6.20(ii), §6.6
Permalink:: http://dlmf.nist.gov/6.11.E2
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6.11.3 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)+i\mathop{\mathrm{f}\/}\nolimits% \!\left(z\right)=\mathop{U\/}\nolimits\!\left(1,1,-iz\right).$

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and : complex variable
Referenced by:: §6.11, §6.20(ii)
Permalink:: http://dlmf.nist.gov/6.11.E3
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© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 6.10 Other Series Expansions 6.12 Asymptotic Expansions