8 Incomplete Gamma and Related Functions Related Functions 8.20 Asymptotic Expansions of $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ 8.22 Mathematical Applications

§8.21 Generalized Sine and Cosine Integrals

Keywords:: cosine integrals, generalized sine and cosine integrals, sine integrals
Permalink:: http://dlmf.nist.gov/8.21

§8.21(i) Definitions: General Values
§8.21(ii) Definitions: Principal Values
§8.21(iii) Integral Representations
§8.21(iv) Interrelations
§8.21(v) Special Values
§8.21(vi) Series Expansions
§8.21(vii) Auxiliary Functions
§8.21(viii) Asymptotic Expansions

§8.21(i) Definitions: General Values

Keywords:: generalized sine and cosine integrals
Referenced by:: §8.21(iv)
Permalink:: http://dlmf.nist.gov/8.21.i

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 $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\pm i\mathop{\mathrm{si}\/}% \nolimits\!\left(a,z\right)=e^{{\pm\frac{1}{2}\pi ia}}\mathop{\Gamma\/}% \nolimits\!\left(a,ze^{{\mp\frac{1}{2}\pi i}}\right),$

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:: : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable and : parameter
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.E1
Encodings:: TeX, pMML, png

8.21.2 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)\pm i\mathop{\mathrm{Si}\/}% \nolimits\!\left(a,z\right)=e^{{\pm\frac{1}{2}\pi ia}}\mathop{\gamma\/}% \nolimits\!\left(a,ze^{{\mp\frac{1}{2}\pi i}}\right).$

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:: : base of exponential function, $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, : complex variable and : 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 with branch point at . Furthermore, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ and $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ are entire functions of , and $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ and $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ are meromorphic functions of with simple poles at $a=-1,-3,-5,\dots$ and $a=0,-2,-4,\dots$ , respectively.

§8.21(ii) Definitions: Principal Values

Keywords:: generalized sine and cosine integrals, principal values
Referenced by:: §8.21(iii)
Permalink:: http://dlmf.nist.gov/8.21.ii

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 $\int_{0}^{\infty}t^{{a-1}}e^{{\pm it}}dt=e^{{\pm\frac{1}{2}\pi ia}}\mathop{% \Gamma\/}\nolimits\!\left(a\right),$ $0<\realpart{a}<1$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : parameter
Referenced by:: §8.21(iv)
Permalink:: http://dlmf.nist.gov/8.21.E3
Encodings:: TeX, pMML, png

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

§8.21(iii) Integral Representations

Notes:: Follow the prescription given in the first paragraph of §8.21(ii). For (8.21.4) and (8.21.5) replace by with $\mathop{\mathrm{ph}\/}\nolimits z=0$ in (8.2.2), deform the path of integration to run along the positive imaginary axis, and replace by . Then extend to the sector $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ by analytic continuation. Similarly for (8.21.6) and (8.21.7).
Keywords:: gamma function, generalized sine and cosine integrals
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.iii

8.21.4 $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)=\int_{z}^{\infty}t^{{a-1}}% \mathop{\sin\/}\nolimits tdt,$ $\realpart{a}<1$ ,

Symbols:: : differential of , $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : parameter
A&S Ref:: 6.5.8 (This function is called 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 $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)=\int_{z}^{\infty}t^{{a-1}}% \mathop{\cos\/}\nolimits tdt,$ $\realpart{a}<1$ ,

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\int$ : integral, $\realpart{}$ : real part, : complex variable and : parameter
A&S Ref:: 6.5.7 (This function is called 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 $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)=\int_{0}^{z}t^{{a-1}}\mathop% {\sin\/}\nolimits tdt,$ $\realpart{a}>-1$ ,

Symbols:: : differential of , $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : parameter
Referenced by:: §8.21(iii), §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E6
Encodings:: TeX, pMML, png

8.21.7 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)=\int_{0}^{z}t^{{a-1}}\mathop% {\cos\/}\nolimits tdt,$ $\realpart{a}>0$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\int$ : integral, $\realpart{}$ : real part, : complex variable and : parameter
Referenced by:: §8.21(iii), §8.21(iv), §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E7
Encodings:: TeX, pMML, png

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

Notes:: Temporarily restrict $0<\realpart{a}<1$ . Then (8.21.8) and (8.21.9) follow immediately from (8.21.3)–(8.21.7). Subsequently, ease the restrictions on by analytic continuation with respect to ; compare §8.21(i).
Keywords:: generalized sine and cosine integrals
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.iv

8.21.8 $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)=\mathop{\Gamma\/}\nolimits\!% \left(a\right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi a\right)-\mathop% {\mathrm{si}\/}\nolimits\!\left(a,z\right),$ $a\neq-1,-3,-5,\dots$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable and : parameter
Referenced by:: §8.21(iv), §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E8
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, : complex variable and : parameter
Referenced by:: §8.21(iv), §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E9
Encodings:: TeX, pMML, png

§8.21(v) Special Values

Notes:: For (8.21.12) and (8.21.13) use (8.21.8) and (8.21.9), and also (8.21.4) and (8.21.5).
Keywords:: generalized sine and cosine integrals
Permalink:: http://dlmf.nist.gov/8.21.v

8.21.10

$\mathop{\mathrm{si}\/}\nolimits\!\left(0,z\right)=-\mathop{\mathrm{si}\/}% \nolimits\!\left(z\right),$

$\mathop{\mathrm{ci}\/}\nolimits\!\left(0,z\right)=-\mathop{\mathrm{Ci}\/}% \nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$ : sine integral and : complex variable
Referenced by:: §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E10
Encodings:: TeX, TeX, pMML, pMML, png, png

8.21.11 $\mathop{\mathrm{Si}\/}\nolimits\!\left(0,z\right)=\mathop{\mathrm{Si}\/}% \nolimits\!\left(z\right).$

Symbols:: $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral and : complex variable
Referenced by:: §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E11
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\sin\/}\nolimits z$ : sine function and : parameter
Referenced by:: §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E12
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral and : parameter
Referenced by:: §8.21(v)
Permalink:: http://dlmf.nist.gov/8.21.E13
Encodings:: TeX, pMML, png

§8.21(vi) Series Expansions

Notes:: (8.21.14) and (8.21.15) are obtained by expansion of the trigonometric functions in (8.21.6), (8.21.7), and termwise integration. See also Luke (1975, p. 115).
Referenced by:: §8.25(i)
Permalink:: http://dlmf.nist.gov/8.21.vi

¶ Power-Series Expansions

Keywords:: generalized sine and cosine integrals

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

Symbols:: : : factorial, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, : complex variable, : parameter and : nonnegative integer
Referenced by:: §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E14
Encodings:: TeX, pMML, png

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

Symbols:: : : factorial, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, : complex variable, : parameter and : nonnegative integer
Referenced by:: §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E15
Encodings:: TeX, pMML, png

¶ Spherical-Bessel-Function Expansions

Keywords:: generalized sine and cosine integrals

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\mathrm{Si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : complex variable, : parameter and : nonnegative integer
Referenced by:: ¶ ‣ §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E16
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : complex variable, : parameter and : nonnegative integer
Referenced by:: ¶ ‣ §8.21(vi)
Permalink:: http://dlmf.nist.gov/8.21.E17
Encodings:: TeX, pMML, png

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

Notes:: (8.21.22) and (8.21.23) follow from (8.21.4), (8.21.5), (8.21.18), and (8.21.19). For (8.21.24) and (8.21.25) assume $\mathop{\mathrm{ph}\/}\nolimits z=0$ , and in the integrals for $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\pm i\mathop{\mathrm{si}\/}% \nolimits\!\left(a,z\right)$ obtained from (8.21.4) and (8.21.5) set $t=(1+\tau)z$ , rotate the integration paths in the $\tau$ -plane through $\pm\frac{1}{2}\pi$ , and apply (8.21.18) and (8.21.19). The restriction $\mathop{\mathrm{ph}\/}\nolimits z=0$ is eased to $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$ by analytic continuation.
Keywords:: generalized sine and cosine integrals
Permalink:: http://dlmf.nist.gov/8.21.vii

8.21.18 $f(a,z)=\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)\mathop{\cos\/}% \nolimits z-\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\mathop{\sin\/}% \nolimits z,$

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, : complex variable, : parameter and : auxiliary function
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 $g(a,z)=\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)\mathop{\sin\/}% \nolimits z+\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)\mathop{\cos\/}% \nolimits z.$

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, : complex variable, : parameter and : auxiliary function
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 $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)=f(a,z)\mathop{\cos\/}% \nolimits z+g(a,z)\mathop{\sin\/}\nolimits z,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{si}\/}\nolimits\!\left(a,z\right)$ : generalized sine integral, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : parameter, : auxiliary function and : auxiliary function
Referenced by:: §8.21(viii)
Permalink:: http://dlmf.nist.gov/8.21.E20
Encodings:: TeX, pMML, png

8.21.21 $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)=-f(a,z)\mathop{\sin\/}% \nolimits z+g(a,z)\mathop{\cos\/}\nolimits z.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{ci}\/}\nolimits\!\left(a,z\right)$ : generalized cosine integral, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : parameter, : auxiliary function and : auxiliary function
Referenced by:: §8.21(viii)
Permalink:: http://dlmf.nist.gov/8.21.E21
Encodings:: TeX, pMML, png

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:: : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, : complex variable, : parameter and : 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, : differential of , $\int$ : integral, : complex variable, : parameter and : 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 $f(a,z)=\frac{z^{a}}{2}\int_{0}^{\infty}\left((1+it)^{{a-1}}+(1-it)^{{a-1}}% \right)e^{{-zt}}dt,$

Symbols:: : differential of , : base of exponential function, $\int$ : integral, : complex variable, : parameter and : auxiliary function
Referenced by:: §8.21(vii), §8.21(viii)
Permalink:: http://dlmf.nist.gov/8.21.E24
Encodings:: TeX, pMML, png

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

Symbols:: : differential of , : base of exponential function, $\int$ : integral, : complex variable, : parameter and : auxiliary function
Referenced by:: §8.21(vii), §8.21(viii)
Permalink:: http://dlmf.nist.gov/8.21.E25
Encodings:: TeX, pMML, png

§8.21(viii) Asymptotic Expansions

Notes:: Apply Watson’s lemma to (8.21.24) and (8.21.25), and then extend the sector of validity from $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi-\delta$ to $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$ ; see §2.4(i).
Keywords:: cosine integrals, generalized sine and cosine integrals, sine integrals
Permalink:: http://dlmf.nist.gov/8.21.viii

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

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\sim$ : asymptotic equality, : complex variable, : parameter, : nonnegative integer and : auxiliary function
Permalink:: http://dlmf.nist.gov/8.21.E26
Encodings:: TeX, pMML, png

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

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\sim$ : asymptotic equality, : complex variable, : parameter, : nonnegative integer and : auxiliary function
Permalink:: http://dlmf.nist.gov/8.21.E27
Encodings:: TeX, pMML, png

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 8.20 Asymptotic Expansions of $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ 8.22 Mathematical Applications