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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.2 Definitions and Interrelations

  1. §6.2(i) Exponential and Logarithmic Integrals
  2. §6.2(ii) Sine and Cosine Integrals
  3. §6.2(iii) Auxiliary Functions

§6.2(i) Exponential and Logarithmic Integrals

The principal value of the exponential integral E1(z) is defined by

6.2.1 E1(z)=zettdt,

where the path does not cross the negative real axis or pass through the origin. As in the case of the logarithm (§4.2(i)) there is a cut along the interval (,0] and the principal value is two-valued on (,0).

Unless indicated otherwise, it is assumed throughout the DLMF that E1(z) assumes its principal value. This is also true of the functions Ci(z) and Chi(z) defined in §6.2(ii).

6.2.2 E1(z)=ez0ett+zdt,
6.2.3 Ein(z)=0z1ettdt.

Ein(z) is sometimes called the complementary exponential integral. It is entire.

6.2.4 E1(z)=Ein(z)lnzγ.

In the next three equations x>0.

6.2.5 Ei(x)=xettdt=xettdt,
6.2.6 Ei(x)=xettdt=E1(x),
6.2.7 Ei(±x)=Ein(x)+lnx+γ.

(Ei(x) is undefined when x=0, or when x is not real.)

The logarithmic integral is defined by

6.2.8 li(x)=0xdtlnt=Ei(lnx),

The generalized exponential integral Ep(z), p, is treated in Chapter 8.

§6.2(ii) Sine and Cosine Integrals

6.2.9 Si(z)=0zsinttdt.

Si(z) is an odd entire function.

6.2.10 si(z)=zsinttdt=Si(z)12π.
6.2.11 Ci(z)=zcosttdt,

where the path does not cross the negative real axis or pass through the origin. This is the principal value; compare (6.2.1).

6.2.12 Cin(z)=0z1costtdt.

Cin(z) is an even entire function.

6.2.13 Ci(z)=Cin(z)+lnz+γ.

Values at Infinity

6.2.14 limxSi(x) =12π,
limxCi(x) =0.

Hyperbolic Analogs of the Sine and Cosine Integrals

6.2.15 Shi(z) =0zsinhttdt,
6.2.16 Chi(z) =γ+lnz+0zcosht1tdt.

§6.2(iii) Auxiliary Functions

6.2.17 f(z) =Ci(z)sinzsi(z)cosz,
6.2.18 g(z) =Ci(z)coszsi(z)sinz.
6.2.19 Si(z) =12πf(z)coszg(z)sinz,
6.2.20 Ci(z) =f(z)sinzg(z)cosz.
6.2.21 df(z)dz =g(z),
dg(z)dz =f(z)1z.