Sine Integral

There are two types of ``sine integrals'' commonly defined,

$$\mathop{\rm Si}\nolimits (x) \equiv \int_0^z {\sin t\over t}\,dt$$ (1)

and

$$\mathop{\rm si}\nolimits (x) \equiv - \int_x^\infty {\sin t\over t}\,dt$$ (2)

$$= {1\over 2i} [\mathop{\rm ei}\nolimits (ix)-\mathop{\rm ei}\nolimits (-ix)]$$

$$= {1\over 2i} [{\rm e}_1(ix)-{\rm e}_1(-ix)]$$ (3)

$$= \mathop{\rm Si}\nolimits (z)-{\textstyle{1\over 2}}\pi,$$ (4)

where $\mathop{\rm ei}\nolimits (x)$ is the Exponential Integral and

$${\rm e}_1(x)\equiv -\mathop{\rm ei}\nolimits (-x).$$ (5)

$\mathop{\rm Si}\nolimits (x)$ is the function returned by the Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Champaign, IL) command SinIntegral[x] and displayed above. The half-infinite integral of the Sinc Function is given by

$$\mathop{\rm si}(0)=-\int_0^\infty {\sin x\over x}\,dx=-{\textstyle{1\over 2}}\pi.$$ (6)

To compute the integral of a sine function times a power

$$I\equiv \int x^{2n}\sin(mx)\,dx,$$ (7)

use Integration by Parts. Let

$$u=x^{2n}\qquad dv=\sin(mx)\,dx$$ (8)

$$du=2nx^{2n-1}\,dx \qquad v=-{1\over m} \cos(mx),$$ (9)

so

$$I=-{1\over m} x^{2n}\cos(mx)+{2n\over m}\int x^{2n-1}\cos(mx)\,dx.$$ (10)

Using Integration by Parts again,

$$u=x^{2n-1}\qquad dv=\cos(mx)\,dx$$ (11)

$$du=(2n-1)x^{2n-2}\,dx \qquad v={1\over m} \sin(mx)$$ (12)

$\int x^{2n}\sin(mx)\,dx =-{1\over m} x^{2n}\cos(mx)$
$\quad \phantom{=} +{2n\over m}\left[{{1\over m} x^{2n-1}\cos(mx) -{2n-1\over m}\int x^{2n-2}\sin(mx)\,dx}\right]$
$\quad = -{1\over m}x^{2n}\sin(mx)+{2n\over m^2} x^{2n-1}\sin(mx) -{(2n)(2n-1)\over m^2} \int x^{2n-2}\sin(mx)\,dx$
$\quad =-{1\over m}x^{2n}\cos(mx)+{2n\over m^2} x^{2n-1}\sin(mx)+\ldots+{(2n)!\over m^{2n}}\int x^0\sin(mx)\,dx$
$\quad =-{1\over m}x^{2n}\cos(mx)+{2n\over m^2} x^{2n-1}\sin(mx)+\ldots -{(2n)!\over m^{2n+1}}\cos(mx)$
$\quad =\cos(mx)\sum_{k=0}^n (-1)^{k+1} {(2n)!\over (2n-2k)! m^{2k+1}} x^{2n-2k}$
$\quad \phantom{=} +\sin(mx)\sum_{k=1}^n (-1)^{k+1} {(2n)!\over (2k-2n-1)!m^{2k}} x^{2n-2k+1}.$	(13)

Letting $k'\equiv n-k$, so

$\int x^{2n}\sin(mx)\,dx$
$\quad =\cos(mx)\sum_{k=0}^n (-1)^{n-k+1} {(2n)!\over (2k)!m^{2n-2k+1}} x^{2k}$
$\quad \phantom{=} + \sin(mx) \sum_{k=0}^{n-1} (-1)^{n-k+1} {(2n)!\over (2k-1)!m^{2n-2k}} x^{2k+1}$
$\quad = (-1)^{n+1}(2n)!\left[{\cos(mx)\sum_{k=0}^n {(-1)^k\over (2k)!m^{2n-2k+1}} x^{2k}}\right.$
$\quad \phantom{=} +\left.{\sin(mx)\sum_{k=1}^n {(-1)^{k+1}\over (2k-3)!m^{2n-2k+2}} x^{2k-1}}\right].$
	(14)

General integrals of the form

$$I(k,l)=\int_0^\infty {\sin^k x\over x^l}\,dx$$ (15)

are related to the Sinc Function and can be computed analytically.

See also Chi, Cosine Integral, Exponential Integral, Nielsen's Spiral, Shi, Sici Spiral, Sinc Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Sine and Cosine Integrals.'' §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231-233, 1972.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342-343, 1985.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Fresnel Integrals, Cosine and Sine Integrals.'' §6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 248-252, 1992.

Spanier, J. and Oldham, K. B. ``The Cosine and Sine Integrals.'' Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361-372, 1987.