§6.5 Further Interrelations

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Keywords:: cosine integrals, exponential integrals, interrelations, relations to exponential integrals, relations to other functions, sine and cosine integrals, sine integrals
Notes:: For (6.5.1) and (6.5.2) see Olver (1997b, p. 41). (6.5.3) and (6.5.4) follow from (6.6.1), (6.6.2), (6.6.5), and (6.6.6). For (6.5.5) and (6.5.6) see Olver (1997b, p. 42). (6.5.7) follows from (6.2.10), (6.2.17), (6.2.18), (6.5.5), and (6.5.6).
Permalink:: http://dlmf.nist.gov/6.5
See also:: Annotations for Ch.6

When $x>0$ ,

6.5.1		$E_{1}\left(-x\pm i0\right)=-\operatorname{Ei}\left(x\right)\mp i\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit and $x$ : real variable A&S Ref: 5.1.7 Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.5.E1 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6

6.5.2		$\operatorname{Ei}\left(x\right)=-\tfrac{1}{2}(E_{1}\left(-x+i0\right)+E_{1}% \left(-x-i0\right)),$
ⓘ Symbols: $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit and $x$ : real variable A&S Ref: 5.1.7 Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.5.E2 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6

6.5.3		$\tfrac{1}{2}(\operatorname{Ei}\left(x\right)+E_{1}\left(x\right))=% \operatorname{Shi}\left(x\right)=-i\operatorname{Si}\left(ix\right),$
ⓘ Symbols: $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\operatorname{Shi}\left(\NVar{z}\right)$ : hyperbolic sine integral, $\mathrm{i}$ : imaginary unit, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $x$ : real variable A&S Ref: 5.2.22 (in modified form) Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.5.E3 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6

6.5.4		$\tfrac{1}{2}(\operatorname{Ei}\left(x\right)-E_{1}\left(x\right))=% \operatorname{Chi}\left(x\right)=\operatorname{Ci}\left(ix\right)-\tfrac{1}{2}% \pi i.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\operatorname{Chi}\left(\NVar{z}\right)$ : hyperbolic cosine integral, $\mathrm{i}$ : imaginary unit and $x$ : real variable A&S Ref: 5.2.24 (in modified form) Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.5.E4 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6

When $|\operatorname{ph}z|<\frac{1}{2}\pi$ ,

6.5.5	$\displaystyle\operatorname{Si}\left(z\right)$	$\displaystyle=\tfrac{1}{2}i(E_{1}\left(-iz\right)-E_{1}\left(iz\right))+\tfrac% {1}{2}\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable A&S Ref: 5.2.21 Referenced by: §6.12(ii), §6.5 Permalink: http://dlmf.nist.gov/6.5.E5 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6
6.5.6	$\displaystyle\operatorname{Ci}\left(z\right)$	$\displaystyle=-\tfrac{1}{2}(E_{1}\left(iz\right)+E_{1}\left(-iz\right)),$
ⓘ Symbols: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit and $z$ : complex variable A&S Ref: 5.2.23 Referenced by: §6.12(ii), §6.5 Permalink: http://dlmf.nist.gov/6.5.E6 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6

6.5.7		$\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=E_{1}\left(\mp iz\right)% e^{\mp iz}.$
ⓘ Symbols: $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : 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, §6.5, §6.7(iii) Permalink: http://dlmf.nist.gov/6.5.E7 Encodings: TeX, pMML, png See also: Annotations for §6.5 and Ch.6