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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.4 Analytic Continuation

Analytic continuation of the principal value of E1(z) yields a multi-valued function with branch points at z=0 and z=. The general value of E1(z) is given by

6.4.1 E1(z)=Ein(z)-Lnz-γ;

compare (6.2.4) and (4.2.6). Thus

6.4.2 E1(ze2mπi)=E1(z)-2mπi,


6.4.3 E1(ze±πi)=Ein(-z)-lnz-γπi,

The general values of the other functions are defined in a similar manner, and

6.4.4 Ci(ze±πi) =±πi+Ci(z),
6.4.5 Chi(ze±πi) =±πi+Chi(z),
6.4.6 f(ze±πi) =πeiz-f(z),
6.4.7 g(ze±πi) =πieiz+g(z).

Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions E1(z), Ci(z), Chi(z), f(z), and g(z) assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.