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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.6 Power Series

6.6.1 Ei(x)=γ+lnx+n=1xnn!n,
6.6.2 E1(z)=-γ-lnz-n=1(-1)nznn!n.
6.6.3 E1(z)=-lnz+e-zn=0znn!ψ(n+1),

where ψ denotes the logarithmic derivative of the gamma function (§5.2(i)).

6.6.4 Ein(z)=n=1(-1)n-1znn!n,
6.6.5 Si(z)=n=0(-1)nz2n+1(2n+1)!(2n+1),
6.6.6 Ci(z)=γ+lnz+n=1(-1)nz2n(2n)!(2n).

The series in this section converge for all finite values of x and |z|.