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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.5 Further Interrelations

When x>0,

6.5.1 E1(-x±i0)=-Ei(x)iπ,
6.5.2 Ei(x)=-12(E1(-x+i0)+E1(-x-i0)),
6.5.3 12(Ei(x)+E1(x))=Shi(x)=-iSi(ix),
6.5.4 12(Ei(x)-E1(x))=Chi(x)=Ci(ix)-12πi.

When |phz|<12π,

6.5.5 Si(z) =12i(E1(-iz)-E1(iz))+12π,
6.5.6 Ci(z) =-12(E1(iz)+E1(-iz)),
6.5.7 g(z)±if(z)=E1(iz)eiz.