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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.10 Other Series Expansions

Contents

§6.10(i) Inverse Factorial Series

6.10.1 E1(z)=-z(c0z+c1z(z+1)+2!c2z(z+1)(z+2)+3!c3z(z+1)(z+2)(z+3)+),
z>0,

where

6.10.2 c0 =1,
c1 =-1,
c2 =12,
c3 =-13,
c4 =16,

and

6.10.3 ck=-j=0k-1cjk-j,
k1.

For a more general result (incomplete gamma function), and also for a result for the logarithmic integral, see Nielsen (1906a, p. 283: Formula (3) is incorrect).

§6.10(ii) Expansions in Series of Spherical Bessel Functions

For the notation see §10.47(ii).

6.10.4 Si(z) =zn=0(jn(12z))2,
6.10.5 Cin(z) =n=1an(jn(12z))2,
6.10.6 Ei(x)=γ+ln|x|+n=0(-1)n(x-an)(in(1)(12x))2,
x0,

where

6.10.7 an=(2n+1)(1-(-1)n+ψ(n+1)-ψ(1)),

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

6.10.8 Ein(z)=z-z/2(i0(1)(12z)+n=12n+1n(n+1)in(1)(12z)).

For (6.10.4)–(6.10.8) and further results see Harris (2000) and Luke (1969b, pp. 56–57). An expansion for E1(z) can be obtained by combining (6.2.4) and (6.10.8).