ABSTRACT
This is easily proved by transforming variables when z is positive, and extending to Jphzl< -$n by analytic continuation.
Ein (z) = 6 dt, (3.03) and is entire. By expanding the integrand in ascending powers o f t and integrating term by term we obtain the Maclaurin expansion
The connection between El(z) and Ein(z) is found by temporarily supposing that z > 0 and rearranging (3.03) in the form
Combination with (3.04) then yields
Analytic continuation immediately extends (3.05) and (3.06) to complex z. In both cases principal branches of El@) and In z correspond. 3.2 When z = x and is real, another notation often used for the exponential integral is given by
it being understood that the integral takes its Cauchy principal value when x is positive.' The connection with the previous notation is given by
with x > 0 in both relations. These identities are obtained by replacing t by - t and, in the case of the second one, using a contour with a vanishingly small indentation. The notation El(-x+iO), for example, means the value of the principal branch of El(-x) on the upper side of the cut.
