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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.13 Zeros

The function Ei(x) has one real zero x0, given by

6.13.1 x0=0.37250 74107 81366 63446 19918 66580.

Ci(x) and si(x) each have an infinite number of positive real zeros, which are denoted by ck, sk, respectively, arranged in ascending order of absolute value for k=0,1,2,. Values of c1 and c2 to 30D are given by MacLeod (1996b).

As k,

6.13.2 ck,skα+1α-1631α3+1673151α5-5 077461051α7+,

where α=kπ for ck, and α=(k+12)π for sk. For these results, together with the next three terms in (6.13.2), see MacLeod (2002a). See also Riekstynš (1991, pp. 176–177).