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8 Incomplete Gamma and Related FunctionsRelated Functions

§8.21 Generalized Sine and Cosine Integrals

Contents

§8.21(i) Definitions: General Values

With γ and Γ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define

8.21.1 ci(a,z)±isi(a,z) =e±12πiaΓ(a,ze12πi),
8.21.2 Ci(a,z)±iSi(a,z) =e±12πiaγ(a,ze12πi).

From §§8.2(i) and 8.2(ii) it follows that each of the four functions si(a,z), ci(a,z), Si(a,z), and Ci(a,z) is a multivalued function of z with branch point at z=0. Furthermore, si(a,z) and ci(a,z) are entire functions of a, and Si(a,z) and Ci(a,z) are meromorphic functions of a with simple poles at a=-1,-3,-5, and a=0,-2,-4,, respectively.

§8.21(ii) Definitions: Principal Values

When phz=0 (and when a-1,-3,-5,, in the case of Si(a,z), or a0,-2,-4,, in the case of Ci(a,z)) the principal values of si(a,z), ci(a,z), Si(a,z), and Ci(a,z) are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). Elsewhere in the sector |phz|π the principal values are defined by analytic continuation from phz=0; compare §4.2(i).

From here on it is assumed that unless indicated otherwise the functions si(a,z), ci(a,z), Si(a,z), and Ci(a,z) have their principal values.

Properties of the four functions that are stated below in §§8.21(iii) and 8.21(iv) follow directly from the definitions given above, together with properties of the incomplete gamma functions given earlier in this chapter. In the case of §8.21(iv) the equation

8.21.3 0ta-1e±itdt=e±12πiaΓ(a),
0<a<1,

(obtained from (5.2.1) by rotation of the integration path) is also needed.

§8.21(iii) Integral Representations

8.21.4 si(a,z) =zta-1sintdt,
a<1,
8.21.5 ci(a,z) =zta-1costdt,
a<1,
8.21.6 Si(a,z) =0zta-1sintdt,
a>-1,
8.21.7 Ci(a,z) =0zta-1costdt,
a>0.

In these representations the integration paths do not cross the negative real axis, and in the case of (8.21.4) and (8.21.5) the paths also exclude the origin.

§8.21(iv) Interrelations

8.21.8 Si(a,z)=Γ(a)sin(12πa)-si(a,z),
a-1,-3,-5,,
8.21.9 Ci(a,z)=Γ(a)cos(12πa)-ci(a,z),
a0,-2,-4,.

§8.21(v) Special Values

8.21.10 si(0,z) =-si(z),
ci(0,z) =-Ci(z),
8.21.11 Si(0,z)=Si(z).

For the functions on the right-hand sides of (8.21.10) and (8.21.11) see §6.2(ii).

8.21.12 Si(a,) =Γ(a)sin(12πa),
a-1,-3,-5,,
8.21.13 Ci(a,) =Γ(a)cos(12πa),
a0,-2,-4,.

§8.21(vi) Series Expansions

Power-Series Expansions

8.21.14 Si(a,z)=zak=0(-1)kz2k+1(2k+a+1)(2k+1)!,
a-1,-3,-5,,
8.21.15 Ci(a,z)=zak=0(-1)kz2k(2k+a)(2k)!,
a0,-2,-4,.

Spherical-Bessel-Function Expansions

8.21.16 Si(a,z) =zak=0(2k+32)(1-12a)k(12+12a)k+1j2k+1(z),
a-1,-3,-5,,
8.21.17 Ci(a,z) =zak=0(2k+12)(12-12a)k(12a)k+1j2k(z),
a0,-2,-4,.

For jn(z) see §10.47(ii). For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57).

§8.21(vii) Auxiliary Functions

8.21.18 f(a,z) =si(a,z)cosz-ci(a,z)sinz,
8.21.19 g(a,z) =si(a,z)sinz+ci(a,z)cosz.
8.21.20 si(a,z) =f(a,z)cosz+g(a,z)sinz,
8.21.21 ci(a,z) =-f(a,z)sinz+g(a,z)cosz.

When |phz|<π and a<1,

8.21.22 f(a,z)=0sint(t+z)1-adt,
8.21.23 g(a,z)=0cost(t+z)1-adt.

When |phz|<12π,

8.21.24 f(a,z)=za20((1+it)a-1+(1-it)a-1)e-ztdt,
8.21.25 g(a,z)=za2i0((1-it)a-1-(1+it)a-1)e-ztdt.

§8.21(viii) Asymptotic Expansions

When z with |phz|π-δ (<π),

8.21.26 f(a,z) za-1k=0(-1)k(1-a)2kz2k,
8.21.27 g(a,z) za-1k=0(-1)k(1-a)2k+1z2k+1.

For the corresponding expansions for si(a,z) and ci(a,z) apply (8.21.20) and (8.21.21).