# §8.21 Generalized Sine and Cosine Integrals

## §8.21(i) Definitions: General Values

With $\mathop{\gamma\/}\nolimits$ and $\mathop{\Gamma\/}\nolimits$ denoting here the general values of the incomplete gamma functions (§8.2(i)), we define

 8.21.1 $\displaystyle\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\pm i\mathop{% \mathrm{si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=e^{\pm\frac{1}{2}\pi ia}\mathop{\Gamma\/}\nolimits\!\left(a,ze^{% \mp\frac{1}{2}\pi i}\right),$ Defines: $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$: generalized cosine integral and $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral Symbols: $e$: base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.21(ii) Permalink: http://dlmf.nist.gov/8.21.E1 Encodings: TeX, pMML, png 8.21.2 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)\pm i\mathop{% \mathrm{Si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=e^{\pm\frac{1}{2}\pi ia}\mathop{\gamma\/}\nolimits\!\left(a,ze^{% \mp\frac{1}{2}\pi i}\right).$ Defines: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$: generalized cosine integral and $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral Symbols: $e$: base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$: incomplete gamma function, $z$: complex variable and $a$: parameter Referenced by: §8.21(ii) Permalink: http://dlmf.nist.gov/8.21.E2 Encodings: TeX, pMML, png

From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$, and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ is a multivalued function of $z$ with branch point at $z=0$. Furthermore, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ and $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ are entire functions of $a$, and $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\dots$ and $a=0,-2,-4,\dots$, respectively.

## §8.21(ii) Definitions: Principal Values

When $\mathop{\mathrm{ph}\/}\nolimits z=0$ (and when $a\neq-1,-3,-5,\dots$, in the case of $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$, or $a\neq 0,-2,-4,\dots$, in the case of $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$) the principal values of $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$, and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ 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 $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ the principal values are defined by analytic continuation from $\mathop{\mathrm{ph}\/}\nolimits z=0$; compare §4.2(i).

From here on it is assumed that unless indicated otherwise the functions $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$, and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ 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 $\int_{0}^{\infty}t^{a-1}e^{\pm it}dt=e^{\pm\frac{1}{2}\pi ia}\mathop{\Gamma\/}% \nolimits\!\left(a\right),$ $0<\realpart{a}<1$,

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

## §8.21(iii) Integral Representations

 8.21.4 $\displaystyle\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{z}^{\infty}t^{a-1}\mathop{\sin\/}\nolimits tdt,$ $\realpart{a}<1$, Symbols: $dx$: differential of $x$, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral, $\int$: integral, $\realpart{}$: real part, $\mathop{\sin\/}\nolimits z$: sine function, $z$: complex variable and $a$: parameter A&S Ref: 6.5.8 (This function is called $S(z,a)$ in AMS 55.) Referenced by: §8.21(iii), §8.21(iii), §8.21(v), §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E4 Encodings: TeX, pMML, png 8.21.5 $\displaystyle\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{z}^{\infty}t^{a-1}\mathop{\cos\/}\nolimits tdt,$ $\realpart{a}<1$, Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $dx$: differential of $x$, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$: generalized cosine integral, $\int$: integral, $\realpart{}$: real part, $z$: complex variable and $a$: parameter A&S Ref: 6.5.7 (This function is called $C(z,a)$ in AMS 55.) Referenced by: §8.21(iii), §8.21(iii), §8.21(v), §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E5 Encodings: TeX, pMML, png 8.21.6 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{0}^{z}t^{a-1}\mathop{\sin\/}\nolimits tdt,$ $\realpart{a}>-1$, 8.21.7 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=\int_{0}^{z}t^{a-1}\mathop{\cos\/}\nolimits tdt,$ $\realpart{a}>0$.

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 $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!% \left(a\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi a\right)-\mathop% {\mathrm{si}\/}\nolimits\!\left(a,z\right),$ $a\neq-1,-3,-5,\dots$,
 8.21.9 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!% \left(a\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi a\right)-\mathop% {\mathrm{ci}\/}\nolimits\!\left(a,z\right),$ $a\neq 0,-2,-4,\dots$.

## §8.21(v) Special Values

 8.21.10 $\displaystyle\mathop{\mathrm{si}\/}\nolimits\!\left(0,z\right)$ $\displaystyle=-\mathop{\mathrm{si}\/}\nolimits\!\left(z\right),$ $\displaystyle\mathop{\mathrm{ci}\/}\nolimits\!\left(0,z\right)$ $\displaystyle=-\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right),$
 8.21.11 $\mathop{\mathrm{Si}\/}\nolimits\!\left(0,z\right)=\mathop{\mathrm{Si}\/}% \nolimits\!\left(z\right).$

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

 8.21.12 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(a,\infty\right)$ $\displaystyle=\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\sin\/}% \nolimits\!\left(\tfrac{1}{2}\pi a\right),$ $a\neq-1,-3,-5,\dots$, 8.21.13 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,\infty\right)$ $\displaystyle=\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\cos\/}% \nolimits\!\left(\tfrac{1}{2}\pi a\right),$ $a\neq 0,-2,-4,\dots$.

## §8.21(vi) Series Expansions

### Power-Series Expansions

 8.21.14 $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}% \frac{(-1)^{k}z^{2k+1}}{(2k+a+1)(2k+1)!},$ $a\neq-1,-3,-5,\dots$,
 8.21.15 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}% \frac{(-1)^{k}z^{2k}}{(2k+a)(2k)!},$ $a\neq 0,-2,-4,\dots$.

### Spherical-Bessel-Function Expansions

 8.21.16 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac{3}{2}\right)\left(1% -\frac{1}{2}a\right)_{k}}{\left(\frac{1}{2}+\frac{1}{2}a\right)_{k+1}}\mathop{% \mathsf{j}_{2k+1}\/}\nolimits\!\left(z\right),$ $a\neq-1,-3,-5,\dots$, 8.21.17 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac{1}{2}\right)\left(% \frac{1}{2}-\frac{1}{2}a\right)_{k}}{\left(\frac{1}{2}a\right)_{k+1}}\mathop{% \mathsf{j}_{2k}\/}\nolimits\!\left(z\right),$ $a\neq 0,-2,-4,\dots$.

For $\mathop{\mathsf{j}_{n}\/}\nolimits\!\left(z\right)$ 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 $\displaystyle f(a,z)$ $\displaystyle=\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)\mathop{\cos\/}% \nolimits z-\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\mathop{\sin\/}% \nolimits z,$ Defines: $f(a,z)$: auxiliary function (locally) Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$: generalized cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral, $\mathop{\sin\/}\nolimits z$: sine function, $z$: complex variable and $a$: parameter A&S Ref: 5.2.6 (This generalizes the form in AMS 55.) Referenced by: §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E18 Encodings: TeX, pMML, png 8.21.19 $\displaystyle g(a,z)$ $\displaystyle=\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)\mathop{\sin\/}% \nolimits z+\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\mathop{\cos\/}% \nolimits z.$ Defines: $g(a,z)$: auxiliary function (locally) Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$: generalized cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$: generalized sine integral, $\mathop{\sin\/}\nolimits z$: sine function, $z$: complex variable and $a$: parameter A&S Ref: 5.2.7 (This generalizes the form in AMS 55.) Referenced by: §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E19 Encodings: TeX, pMML, png 8.21.20 $\displaystyle\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=f(a,z)\mathop{\cos\/}\nolimits z+g(a,z)\mathop{\sin\/}\nolimits z,$ 8.21.21 $\displaystyle\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ $\displaystyle=-f(a,z)\mathop{\sin\/}\nolimits z+g(a,z)\mathop{\cos\/}\nolimits z.$

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ and $\realpart{a}<1$,

 8.21.22 $f(a,z)=\int_{0}^{\infty}\frac{\mathop{\sin\/}\nolimits t}{(t+z)^{1-a}}dt,$ Symbols: $dx$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits z$: sine function, $z$: complex variable, $a$: parameter and $f(a,z)$: auxiliary function A&S Ref: 5.2.12 (This generalizes the form in AMS 55.) Referenced by: §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E22 Encodings: TeX, pMML, png
 8.21.23 $g(a,z)=\int_{0}^{\infty}\frac{\mathop{\cos\/}\nolimits t}{(t+z)^{1-a}}dt.$ Symbols: $\mathop{\cos\/}\nolimits z$: cosine function, $dx$: differential of $x$, $\int$: integral, $z$: complex variable, $a$: parameter and $g(a,z)$: auxiliary function A&S Ref: 5.2.13 (This generalizes the form in AMS 55.) Referenced by: §8.21(vii) Permalink: http://dlmf.nist.gov/8.21.E23 Encodings: TeX, pMML, png

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$,

 8.21.24 $f(a,z)=\frac{z^{a}}{2}\int_{0}^{\infty}\left((1+it)^{a-1}+(1-it)^{a-1}\right)e% ^{-zt}dt,$
 8.21.25 $g(a,z)=\frac{z^{a}}{2i}\int_{0}^{\infty}\left((1-it)^{a-1}-(1+it)^{a-1}\right)% e^{-zt}dt.$

## §8.21(viii) Asymptotic Expansions

When $z\to\infty$ with $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ($<\pi$),

 8.21.26 $\displaystyle f(a,z)$ $\displaystyle\sim z^{a-1}\sum_{k=0}^{\infty}\frac{(-1)^{k}\left(1-a\right)_{2k% }}{z^{2k}},$ 8.21.27 $\displaystyle g(a,z)$ $\displaystyle\sim z^{a-1}\sum_{k=0}^{\infty}\frac{(-1)^{k}\left(1-a\right)_{2k% +1}}{z^{2k+1}}.$

For the corresponding expansions for $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ and $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ apply (8.21.20) and (8.21.21).