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Sine Integral

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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

si(x)=-int_x^infty(sint)/tdt
(2)
=1/(2i)[Ei(ix)-Ei(-ix)]
(3)
=1/(2i)[e_1(ix)-e_1(-ix)]
(4)
=Si(z)-1/2pi,
(5)

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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