# §6.5 Further Interrelations

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 $\tfrac{1}{2}(\operatorname{Ei}\left(x\right)-E_{1}\left(x\right))=% \operatorname{Chi}\left(x\right)=\operatorname{Ci}\left(ix\right)-\tfrac{1}{2}% \pi i.$

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 $\mathrm{g}\left(z\right)\pm i\mathrm{f}\left(z\right)=E_{1}\left(\mp iz\right)% e^{\mp iz}.$