§6.4 Analytic Continuation

ⓘ

Keywords:: analytic continuation, auxiliary functions, auxiliary functions for sine and cosine integrals, cosine integrals, exponential integrals, hyperbolic analog, principal values
Notes:: For (6.4.1) see Olver (1997b, p. 40). (6.4.3) follows from (6.6.2) and (6.6.4). (6.4.4) and (6.4.5) follow from (6.2.13) and (6.2.16). (6.4.6) and (6.4.7) follow from (6.2.17), (6.2.18), and (6.4.4).
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.4
See also:: Annotations for Ch.6

Analytic continuation of the principal value of $E_{1}\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$ . The general value of $E_{1}\left(z\right)$ is given by

6.4.1		$E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\operatorname{Ln}z-\gamma;$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function and $z$ : complex variable Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.4.E1 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6

compare (6.2.4) and (4.2.6). Thus

6.4.2		$E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right)-2m\pi i,$
$m\in\mathbb{Z}$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers and $z$ : complex variable Permalink: http://dlmf.nist.gov/6.4.E2 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6

and

6.4.3		$E_{1}\left(ze^{\pm\pi i}\right)=\operatorname{Ein}\left(-z\right)-\ln z-\gamma% \mp\pi i,$
$\|\operatorname{ph}z\|\leq\pi$ .
ⓘ Symbols: $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase and $z$ : complex variable Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.4.E3 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6

The general values of the other functions are defined in a similar manner, and

6.4.4	$\displaystyle\operatorname{Ci}\left(ze^{\pm\pi i}\right)$	$\displaystyle=\pm\pi i+\operatorname{Ci}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.4.E4 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6
6.4.5	$\displaystyle\operatorname{Chi}\left(ze^{\pm\pi i}\right)$	$\displaystyle=\pm\pi i+\operatorname{Chi}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{Chi}\left(\NVar{z}\right)$ : hyperbolic cosine integral, $\mathrm{i}$ : imaginary unit and $z$ : complex variable Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.4.E5 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6
6.4.6	$\displaystyle\mathrm{f}\left(ze^{\pm\pi i}\right)$	$\displaystyle=\pi e^{\mp iz}-\mathrm{f}\left(z\right),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.12(ii), §6.4 Permalink: http://dlmf.nist.gov/6.4.E6 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6
6.4.7	$\displaystyle\mathrm{g}\left(ze^{\pm\pi i}\right)$	$\displaystyle=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.12(ii), §6.4 Permalink: http://dlmf.nist.gov/6.4.E7 Encodings: TeX, pMML, png See also: Annotations for §6.4 and Ch.6

Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_{1}\left(z\right)$ , $\operatorname{Ci}\left(z\right)$ , $\operatorname{Chi}\left(z\right)$ , $\mathrm{f}\left(z\right)$ , and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.