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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.2 Definitions and Interrelations


§6.2(i) Exponential and Logarithmic Integrals

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

6.2.1 E1(z)=ze-ttdt,

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)=e-z0e-tt+zdt,
6.2.3 Ein(z)=0z1-e-ttdt.

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

6.2.4 E1(z)=Ein(z)-lnz-γ.

In the next three equations x>0.

6.2.5 Ei(x)=--xe-ttdt=-xettdt,
6.2.6 Ei(-x)=-xe-ttdt=-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)=0z1-costtdt.

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+0zcosht-1tdt.

§6.2(iii) Auxiliary Functions

6.2.17 f(z) =Ci(z)sinz-si(z)cosz,
6.2.18 g(z) =-Ci(z)cosz-si(z)sinz.
6.2.19 Si(z) =12π-f(z)cosz-g(z)sinz,
6.2.20 Ci(z) =f(z)sinz-g(z)cosz.
6.2.21 df(z)dz =-g(z),
dg(z)dz =f(z)-1z.