# §6.5 Further Interrelations

When $x>0$,

 6.5.1 $\mathop{E_{1}\/}\nolimits\!\left(-x\pm i0\right)=-\mathop{\mathrm{Ei}\/}% \nolimits\!\left(x\right)\mp i\pi,$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{E_{1}\/}\nolimits\!\left(\NVar{z}\right)$: exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(\NVar{x}\right)$: exponential integral 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
 6.5.2 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=-\tfrac{1}{2}(\mathop{E_{1}\/}% \nolimits\!\left(-x+i0\right)+\mathop{E_{1}\/}\nolimits\!\left(-x-i0\right)),$ Symbols: $\mathop{E_{1}\/}\nolimits\!\left(\NVar{z}\right)$: exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(\NVar{x}\right)$: exponential integral 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
 6.5.3 $\tfrac{1}{2}(\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)+\mathop{E_{1}\/}% \nolimits\!\left(x\right))=\mathop{\mathrm{Shi}\/}\nolimits\!\left(x\right)=-i% \mathop{\mathrm{Si}\/}\nolimits\!\left(ix\right),$
 6.5.4 $\tfrac{1}{2}(\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)-\mathop{E_{1}\/}% \nolimits\!\left(x\right))=\mathop{\mathrm{Chi}\/}\nolimits\!\left(x\right)=% \mathop{\mathrm{Ci}\/}\nolimits\!\left(ix\right)-\tfrac{1}{2}\pi i.$

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$,

 6.5.5 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ $\displaystyle=\tfrac{1}{2}i(\mathop{E_{1}\/}\nolimits\!\left(-iz\right)-% \mathop{E_{1}\/}\nolimits\!\left(iz\right))+\tfrac{1}{2}\pi,$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{E_{1}\/}\nolimits\!\left(\NVar{z}\right)$: exponential integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: 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 6.5.6 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ $\displaystyle=-\tfrac{1}{2}(\mathop{E_{1}\/}\nolimits\!\left(iz\right)+\mathop% {E_{1}\/}\nolimits\!\left(-iz\right)),$ Symbols: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{E_{1}\/}\nolimits\!\left(\NVar{z}\right)$: exponential integral 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
 6.5.7 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i\mathop{\mathrm{f}\/}% \nolimits\!\left(z\right)=\mathop{E_{1}\/}\nolimits\!\left(\mp iz\right)e^{\mp iz}.$