# §8.21 Generalized Sine and Cosine Integrals

## §8.21(i) Definitions: General Values

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

8.21.1
8.21.2

From §§8.2(i) and 8.2(ii) it follows that each of the four functions , , , and is a multivalued function of with branch point at . Furthermore, and are entire functions of , and and are meromorphic functions of with simple poles at and , respectively.

## §8.21(ii) Definitions: Principal Values

When (and when , in the case of , or , in the case of ) the principal values of , , , and 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 the principal values are defined by analytic continuation from ; compare §4.2(i).

From here on it is assumed that unless indicated otherwise the functions , , , and 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

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

## §8.21(iii) Integral Representations

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(v) Special Values

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

When and ,

8.21.22
8.21.23

## §8.21(viii) Asymptotic Expansions

For the corresponding expansions for and apply (8.21.20) and (8.21.21).