8 Incomplete Gamma and Related Functions Related Functions 8.20 Asymptotic Expansions of

E_{p}\left(z\right)

§8.21 Generalized Sine and Cosine Integrals

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Keywords:: cosine integrals, generalized, generalized sine and cosine integrals, sine integrals
Permalink:: http://dlmf.nist.gov/8.21
See also:: Annotations for Ch.8

§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

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Keywords:: analytic properties, definitions, general values, generalized sine and cosine integrals
Referenced by:: §8.21(iv)
Permalink:: http://dlmf.nist.gov/8.21.i
See also:: Annotations for §8.21 and Ch.8

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

8.21.1	$\displaystyle\operatorname{ci}\left(a,z\right)\pm i\operatorname{si}\left(a,z\right)$	$\displaystyle=e^{\pm\frac{1}{2}\pi ia}\Gamma\left(a,ze^{\mp\frac{1}{2}\pi i}% \right),$
ⓘ Defines: $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral and $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : 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 See also: Annotations for §8.21(i), §8.21 and Ch.8
8.21.2	$\displaystyle\operatorname{Ci}\left(a,z\right)\pm i\operatorname{Si}\left(a,z\right)$	$\displaystyle=e^{\pm\frac{1}{2}\pi ia}\gamma\left(a,ze^{\mp\frac{1}{2}\pi i}% \right).$
ⓘ Defines: $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral and $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : 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 See also: Annotations for §8.21(i), §8.21 and Ch.8

From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\operatorname{si}\left(a,z\right)$ , $\operatorname{ci}\left(a,z\right)$ , $\operatorname{Si}\left(a,z\right)$ , and $\operatorname{Ci}\left(a,z\right)$ is a multivalued function of $z$ with branch point at $z=0$ . Furthermore, $\operatorname{si}\left(a,z\right)$ and $\operatorname{ci}\left(a,z\right)$ are entire functions of $a$ , and $\operatorname{Si}\left(a,z\right)$ and $\operatorname{Ci}\left(a,z\right)$ 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

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Keywords:: definitions, generalized sine and cosine integrals, principal values
Referenced by:: §8.21(iii)
Permalink:: http://dlmf.nist.gov/8.21.ii
See also:: Annotations for §8.21 and Ch.8

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

From here on it is assumed that unless indicated otherwise the functions $\operatorname{si}\left(a,z\right)$ , $\operatorname{ci}\left(a,z\right)$ , $\operatorname{Si}\left(a,z\right)$ , and $\operatorname{Ci}\left(a,z\right)$ 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		$\int_{0}^{\infty}t^{a-1}e^{\pm\mathrm{i}t}\,\mathrm{d}t=e^{\pm\frac{1}{2}\pi% \mathrm{i}a}\Gamma\left(a\right),$
$0<\Re a<1$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $a$ : parameter Referenced by: §8.21(iv) Permalink: http://dlmf.nist.gov/8.21.E3 Encodings: TeX, pMML, png See also: Annotations for §8.21(ii), §8.21 and Ch.8

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

§8.21(iii) Integral Representations

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Keywords:: gamma function, generalized sine and cosine integrals, integral representations
Notes:: Follow the prescription given in the first paragraph of §8.21(ii). For (8.21.4) and (8.21.5) replace $z$ by $\mathrm{i}z$ with $\operatorname{ph}z=0$ in (8.2.2), deform the path of integration to run along the positive imaginary axis, and replace $t$ by $\mathrm{i}t$ . Then extend to the sector $|\operatorname{ph}z|\leq\pi$ by analytic continuation. Similarly for (8.21.6) and (8.21.7).
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.iii
See also:: Annotations for §8.21 and Ch.8

8.21.4	$\displaystyle\operatorname{si}\left(a,z\right)$	$\displaystyle=\int_{z}^{\infty}t^{a-1}\sin t\,\mathrm{d}t,$
$\Re a<1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : 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 See also: Annotations for §8.21(iii), §8.21 and Ch.8
8.21.5	$\displaystyle\operatorname{ci}\left(a,z\right)$	$\displaystyle=\int_{z}^{\infty}t^{a-1}\cos t\,\mathrm{d}t,$
$\Re a<1$ ,
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\int$ : integral, $\Re$ : 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 See also: Annotations for §8.21(iii), §8.21 and Ch.8
8.21.6	$\displaystyle\operatorname{Si}\left(a,z\right)$	$\displaystyle=\int_{0}^{z}t^{a-1}\sin t\,\mathrm{d}t,$
$\Re a>-1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : parameter Referenced by: §8.21(iii), §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E6 Encodings: TeX, pMML, png See also: Annotations for §8.21(iii), §8.21 and Ch.8
8.21.7	$\displaystyle\operatorname{Ci}\left(a,z\right)$	$\displaystyle=\int_{0}^{z}t^{a-1}\cos t\,\mathrm{d}t,$
$\Re a>0$ .
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter Referenced by: §8.21(iii), §8.21(iv), §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E7 Encodings: TeX, pMML, png See also: Annotations for §8.21(iii), §8.21 and Ch.8

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

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Keywords:: generalized sine and cosine integrals, interrelations
Notes:: Temporarily restrict $0<\Re a<1$ . Then (8.21.8) and (8.21.9) follow immediately from (8.21.3)–(8.21.7). Subsequently, ease the restrictions on $a$ by analytic continuation with respect to $a$ ; compare §8.21(i).
Referenced by:: §8.21(ii)
Permalink:: http://dlmf.nist.gov/8.21.iv
See also:: Annotations for §8.21 and Ch.8

8.21.8		$\operatorname{Si}\left(a,z\right)=\Gamma\left(a\right)\sin\left(\tfrac{1}{2}% \pi a\right)-\operatorname{si}\left(a,z\right),$
$a\neq-1,-3,-5,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $a$ : parameter Referenced by: §8.21(iv), §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E8 Encodings: TeX, pMML, png See also: Annotations for §8.21(iv), §8.21 and Ch.8

8.21.9		$\operatorname{Ci}\left(a,z\right)=\Gamma\left(a\right)\cos\left(\tfrac{1}{2}% \pi a\right)-\operatorname{ci}\left(a,z\right),$
$a\neq 0,-2,-4,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $z$ : complex variable and $a$ : parameter Referenced by: §8.21(iv), §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E9 Encodings: TeX, pMML, png See also: Annotations for §8.21(iv), §8.21 and Ch.8

§8.21(v) Special Values

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Keywords:: generalized sine and cosine integrals, relation to sine and cosine integrals, 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).
Permalink:: http://dlmf.nist.gov/8.21.v
See also:: Annotations for §8.21 and Ch.8

8.21.10	$\displaystyle\operatorname{si}\left(0,z\right)$	$\displaystyle=-\operatorname{si}\left(z\right),$
8.21.10	$\displaystyle\operatorname{ci}\left(0,z\right)$	$\displaystyle=-\operatorname{Ci}\left(z\right),$
ⓘ Symbols: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable Referenced by: §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E10 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §8.21(v), §8.21 and Ch.8

8.21.11		$\operatorname{Si}\left(0,z\right)=\operatorname{Si}\left(z\right).$
ⓘ Symbols: $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable Referenced by: §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E11 Encodings: TeX, pMML, png See also: Annotations for §8.21(v), §8.21 and Ch.8

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

8.21.12	$\displaystyle\operatorname{Si}\left(a,\infty\right)$	$\displaystyle=\Gamma\left(a\right)\sin\left(\tfrac{1}{2}\pi a\right),$
$a\neq-1,-3,-5,\dots$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\sin\NVar{z}$ : sine function and $a$ : parameter Referenced by: §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E12 Encodings: TeX, pMML, png See also: Annotations for §8.21(v), §8.21 and Ch.8
8.21.13	$\displaystyle\operatorname{Ci}\left(a,\infty\right)$	$\displaystyle=\Gamma\left(a\right)\cos\left(\tfrac{1}{2}\pi a\right),$
$a\neq 0,-2,-4,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral and $a$ : parameter Referenced by: §8.21(v) Permalink: http://dlmf.nist.gov/8.21.E13 Encodings: TeX, pMML, png See also: Annotations for §8.21(v), §8.21 and Ch.8

§8.21(vi) Series Expansions

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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
See also:: Annotations for §8.21 and Ch.8

Power-Series Expansions

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Keywords:: generalized sine and cosine integrals, power-series expansions
See also:: Annotations for §8.21(vi), §8.21 and Ch.8

8.21.14		$\operatorname{Si}\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k+% 1}}{(2k+a+1)(2k+1)!},$
$a\neq-1,-3,-5,\dots$ ,
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $z$ : complex variable, $a$ : parameter and $k$ : nonnegative integer Referenced by: §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E14 Encodings: TeX, pMML, png See also: Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

8.21.15		$\operatorname{Ci}\left(a,z\right)=z^{a}\sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k}% }{(2k+a)(2k)!},$
$a\neq 0,-2,-4,\dots$ .
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $z$ : complex variable, $a$ : parameter and $k$ : nonnegative integer Referenced by: §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E15 Encodings: TeX, pMML, png See also: Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

Spherical-Bessel-Function Expansions

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Keywords:: expansions in series of spherical Bessel functions, generalized sine and cosine integrals
See also:: Annotations for §8.21(vi), §8.21 and Ch.8

8.21.16	$\displaystyle\operatorname{Si}\left(a,z\right)$	$\displaystyle=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac{3}{2}\right){\left(% 1-\frac{1}{2}a\right)_{k}}}{{\left(\frac{1}{2}+\frac{1}{2}a\right)_{k+1}}}% \mathsf{j}_{2k+1}\left(z\right),$
$a\neq-1,-3,-5,\dots$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\operatorname{Si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $z$ : complex variable, $a$ : parameter and $k$ : nonnegative integer Referenced by: §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E16 Encodings: TeX, pMML, png See also: Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8
8.21.17	$\displaystyle\operatorname{Ci}\left(a,z\right)$	$\displaystyle=z^{a}\sum_{k=0}^{\infty}\frac{\left(2k+\frac{1}{2}\right){\left(% \frac{1}{2}-\frac{1}{2}a\right)_{k}}}{{\left(\frac{1}{2}a\right)_{k+1}}}% \mathsf{j}_{2k}\left(z\right),$
$a\neq 0,-2,-4,\dots$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\operatorname{Ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $z$ : complex variable, $a$ : parameter and $k$ : nonnegative integer Referenced by: §8.21(vi) Permalink: http://dlmf.nist.gov/8.21.E17 Encodings: TeX, pMML, png See also: Annotations for §8.21(vi), §8.21(vi), §8.21 and Ch.8

For $\mathsf{j}_{n}\left(z\right)$ 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

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Keywords:: auxiliary functions, generalized sine and cosine integrals, integral representations
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 $\operatorname{ph}z=0$ , and in the integrals for $\operatorname{ci}\left(a,z\right)\pm i\operatorname{si}\left(a,z\right)$ 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 $\operatorname{ph}z=0$ is eased to $|\operatorname{ph}z|<\frac{1}{2}\pi$ by analytic continuation.
Permalink:: http://dlmf.nist.gov/8.21.vii
See also:: Annotations for §8.21 and Ch.8

8.21.18	$\displaystyle f(a,z)$	$\displaystyle=\operatorname{si}\left(a,z\right)\cos z-\operatorname{ci}\left(a% ,z\right)\sin z,$
ⓘ Defines: $f(a,z)$ : auxiliary function (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\sin\NVar{z}$ : 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 See also: Annotations for §8.21(vii), §8.21 and Ch.8
8.21.19	$\displaystyle g(a,z)$	$\displaystyle=\operatorname{si}\left(a,z\right)\sin z+\operatorname{ci}\left(a% ,z\right)\cos z.$
ⓘ Defines: $g(a,z)$ : auxiliary function (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\sin\NVar{z}$ : 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 See also: Annotations for §8.21(vii), §8.21 and Ch.8
8.21.20	$\displaystyle\operatorname{si}\left(a,z\right)$	$\displaystyle=f(a,z)\cos z+g(a,z)\sin z,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{si}\left(\NVar{a},\NVar{z}\right)$ : generalized sine integral, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter, $f(a,z)$ : auxiliary function and $g(a,z)$ : auxiliary function Referenced by: §8.21(viii) Permalink: http://dlmf.nist.gov/8.21.E20 Encodings: TeX, pMML, png See also: Annotations for §8.21(vii), §8.21 and Ch.8
8.21.21	$\displaystyle\operatorname{ci}\left(a,z\right)$	$\displaystyle=-f(a,z)\sin z+g(a,z)\cos z.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{ci}\left(\NVar{a},\NVar{z}\right)$ : generalized cosine integral, $\sin\NVar{z}$ : sine function, $z$ : complex variable, $a$ : parameter, $f(a,z)$ : auxiliary function and $g(a,z)$ : auxiliary function Referenced by: §8.21(viii) Permalink: http://dlmf.nist.gov/8.21.E21 Encodings: TeX, pMML, png See also: Annotations for §8.21(vii), §8.21 and Ch.8

When $|\operatorname{ph}z|<\pi$ and $\Re a<1$ ,

8.21.22		$f(a,z)=\int_{0}^{\infty}\frac{\sin t}{(t+z)^{1-a}}\,\mathrm{d}t,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : 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 See also: Annotations for §8.21(vii), §8.21 and Ch.8

8.21.23		$g(a,z)=\int_{0}^{\infty}\frac{\cos t}{(t+z)^{1-a}}\,\mathrm{d}t.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : 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 See also: Annotations for §8.21(vii), §8.21 and Ch.8

When $|\operatorname{ph}z|<\frac{1}{2}\pi$ ,

8.21.24		$f(a,z)=\frac{z^{a}}{2}\int_{0}^{\infty}\left((1+it)^{a-1}+(1-it)^{a-1}\right)e% ^{-zt}\,\mathrm{d}t,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $a$ : parameter and $f(a,z)$ : auxiliary function Referenced by: §8.21(vii), §8.21(viii) Permalink: http://dlmf.nist.gov/8.21.E24 Encodings: TeX, pMML, png See also: Annotations for §8.21(vii), §8.21 and Ch.8

8.21.25		$g(a,z)=\frac{z^{a}}{2i}\int_{0}^{\infty}\left((1-it)^{a-1}-(1+it)^{a-1}\right)% e^{-zt}\,\mathrm{d}t.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $a$ : parameter and $g(a,z)$ : auxiliary function Referenced by: §8.21(vii), §8.21(viii) Permalink: http://dlmf.nist.gov/8.21.E25 Encodings: TeX, pMML, png See also: Annotations for §8.21(vii), §8.21 and Ch.8

§8.21(viii) Asymptotic Expansions

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Keywords:: asymptotic expansions for large variable, auxiliary functions, cosine integrals, generalized, generalized sine and cosine integrals, sine integrals
Notes:: Apply Watson’s lemma to (8.21.24) and (8.21.25), and then extend the sector of validity from $|\operatorname{ph}z|\leq\frac{1}{2}\pi-\delta$ to $|\operatorname{ph}z|\leq\pi-\delta$ ; see §2.4(i).
Permalink:: http://dlmf.nist.gov/8.21.viii
See also:: Annotations for §8.21 and Ch.8

When $z\to\infty$ with $|\operatorname{ph}z|\leq\pi-\delta$ ( $<\pi$ ),

8.21.26	$\displaystyle f(a,z)$	$\displaystyle\sim z^{a-1}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(1-a\right)_{2% k}}}{z^{2k}},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $f(a,z)$ : auxiliary function Permalink: http://dlmf.nist.gov/8.21.E26 Encodings: TeX, pMML, png See also: Annotations for §8.21(viii), §8.21 and Ch.8
8.21.27	$\displaystyle g(a,z)$	$\displaystyle\sim z^{a-1}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(1-a\right)_{2% k+1}}}{z^{2k+1}}.$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer and $g(a,z)$ : auxiliary function Permalink: http://dlmf.nist.gov/8.21.E27 Encodings: TeX, pMML, png See also: Annotations for §8.21(viii), §8.21 and Ch.8