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The most common “sine integral” is defined as

 Si(z)=int_0^z(sint)/tdt

(1)

Si(z) is the function implemented in the
Wolfram Language as the function SinIntegral[z].

Si(z) is an entire
function.

A closed related function is defined by

where Ei(x)
is the exponential integral, (3)
holds for x<0,
and

 e_1(x)=-Ei(-x).

(6)

The derivative of Si(x) is

 d/(dx)Si(x)=sinc(x),

(7)

where sinc(x)
is the sinc function and the integral
is

 intSi(x)dx=cosx+xSi(x).

(8)

A series for Si(x)
is given by

 Si(x)=sum_(k=1)^infty(-1)^(k-1)(x^(2k-1))/((2k-1)(2k-1)!)

(9)

(Havil 2003, p. 106).

It has an expansion in terms of spherical
Bessel functions of the first kind as

 Si(2x)=2xsum_(n=0)^infty[j_n(x)]^2

(10)

(Harris 2000).

The half-infinite integral of the sinc function
is given by

 si(0)=-int_0^infty(sinx)/xdx=-1/2pi.

(11)

To compute the integral of a sine function times a power

 I=intx^(2n)sin(mx)dx,

(12)

use integration by parts. Let

 u=x^(2n) dv=sin(mx)dx

(13)

 du=2nx^(2n-1)dx v=-1/mcos(mx),

(14)

so

 I=-1/mx^(2n)cos(mx)+(2n)/mintx^(2n-1)cos(mx)dx.

(15)

Using integration by parts again,

 u=x^(2n-1) dv=cos(mx)dx

(16)

 du=(2n-1)x^(2n-2)dx v=1/msin(mx)

(17)

 intx^(2n)sin(mx)dx=-1/mx^(2n)cos(mx) +(2n)/m[1/mx^(2n-1)cos(mx)-(2n-1)/mintx^(2n-2)sin(mx)dx] =-1/mx^(2n)sin(mx)+(2n)/(m^2)x^(2n-1)sin(mx)-((2n)(2n-1))/(m^2)intx^(2n-2)sin(mx)dx =-1/mx^(2n)cos(mx)+(2n)/(m^2)x^(2n-1)sin(mx)+...+((2n)!)/(m^(2n))intx^0sin(mx)dx =-1/mx^(2n)cos(mx)+(2n)/(m^2)x^(2n-1)sin(mx)+...-((2n)!)/(m^(2n+1))cos(mx) =cos(mx)sum_(k=0)^n(-1)^(k+1)((2n)!)/((2n-2k)!m^(2k+1))x^(2n-2k) +sin(mx)sum_(k=1)^n(-1)^(k+1)((2n)!)/((2k-2n-1)!m^(2k))x^(2n-2k+1).

(18)

Letting k^'=n-k,
so

 intx^(2n)sin(mx)dx =cos(mx)sum_(k=0)^n(-1)^(n-k+1)((2n)!)/((2k)!m^(2n-2k+1))x^(2k)+sin(mx)sum_(k=0)^(n-1)(-1)^(n-k+1)((2n)!)/((2k-1)!m^(2n-2k))x^(2k+1) =(-1)^(n+1)(2n)![cos(mx)sum_(k=0)^n((-1)^k)/((2k)!m^(2n-2k+1))x^(2k)+sin(mx)sum_(k=1)^n((-1)^(k+1))/((2k-3)!m^(2n-2k+2))x^(2k-1)].

(19)

General integrals of the form

 I(k,l)=int_0^infty(sin^kx)/(x^l)dx

(20)

are related to the sinc function and can be computed
analytically.

See also

Chi

,

Cosine Integral

,

Exponential Integral

,

Nielsen’s
Spiral

,

Shi

,

Sinc Function

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/SinIntegral/

Explore with Wolfram|Alpha

References

Abramowitz, M. and Stegun, I. 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.Harris, F. E. “Spherical Bessel Expansions of Sine, Cosine,
and Exponential Integrals.” Appl. Numer. Math. 34, 95-98, 2000.Havil,
J. Gamma:
Exploring Euler’s Constant. Princeton, NJ: Princeton University Press, pp. 105-106,
2003.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.

Referenced
on Wolfram|Alpha

Sine Integral

Cite this as:

Weisstein, Eric W. “Sine Integral.” From
MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/SineIntegral.html

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