6.4 Analytic Continuation6.6 Power Series

§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,
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)),
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.