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Cosine Integral

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CosIntegral
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The most common form of cosine integral is

Ci(x)=-int_x^infty(costdt)/t
(1)
=gamma+lnx+int_0^x(cost-1)/tdt
(2)
=1/2[Ei(ix)+Ei(-ix)]
(3)
=-1/2[E_1(ix)+E_1(-ix)],
(4)

where Ei(x) is the exponential integral, E_n(x) is the En-function, and gamma is the Euler-Mascheroni constant.

Ci(x) is returned by the Wolfram Language command CosIntegral[x], and is also commonly denoted ci(x).

Ci(x) has the series expansion

Ci(x)=gamma+lnx+sum_(k=1)^infty((-x^2)^k)/(2k(2k)!)
(5)

(Havil 2003, p. 106; after inserting a minus sign in the definition).

The derivative is

d/(dz)Ci(z)=(cosz)/z,
(6)

and the integral is

intCi(z)dz=zCi(z)-sinz.
(7)

Ci(x) has zeros at 0.616505, 3.38418, 6.42705, .... Extrema occur when

Ci^'(x)=(cosx)/x=0,
(8)

or cosx=0, or pi/2, 3pi/2, 5pi/2, ..., which are alternately maxima and minima. At these points, Ci(x) equals 0.472001, -0.198408, 0.123772, .... Inflection points occur when

Ci^('')(x)=-(cosx)/(x^2)-(sinx)/x=0,
(9)

which simplifies to

1+xtanx=0,
(10)

which has solutions 2.79839, 6.12125, 9.31787, ....

The related function

Cin(x)=int_0^z((1-cost)dt)/t
(11)
=-ci(x)+lnx+gamma
(12)

is sometimes also defined.

To find a closed form for an integral power of a cosine function, use integration by parts to obtain

I=intcos^mxdx
(13)
=(sinxcos^(m-1)x)/m+(m-1)/mintcos^(m-2)xdx.
(14)

Now, if m is even so m=2n, then

intcos^(2n)xdx=sinxsum_(k=1)^(n)((2k-2)!!)/((2n)!!)((2n-1)!!)/((2k-1)!!)cos^(2k-1)x+((2n-1)!!)/((2n)!!)x
(15)
=((2n-1)!!)/((2n)!!)[sinxsum_(k=0)^(n-1)((2k)!!)/((2k+1)!!)cos^(2k+1)x+x].
(16)

On the other hand, if m is odd so m=2n+1, then

intcos^(2n+1)xdx=sinxsum_(k=0)^n((2n-2k-1)!!)/((2n+1)!!)((2n)!!)/((2n-2k)!!)cos^(2n-2k)x.
(17)

Now let k^'=n-k,

intcos^(2n)xdx=((2n)!!)/((2n+1)!!)sinxsum_(k=0)^n((2k-1)!!)/((2k)!!)cos^(2k)x.
(18)

The general result is then

intcos^mxdx={((2n-1)!!)/((2n)!!)[sinxsum_(k=0)^(n-1)((2k)!!)/((2k+1)!!)cos^(2k+1)x+x] for m=2n; ((2n)!!)/((2n+1)!!)sinxsum_(k=0)^n((2k-1)!!)/((2k)!!)cos^(2k)x for m=2n+1.
(19)

The infinite integral of a cosine times a Gaussian can also be done in closed form,

int_(-infty)^inftye^(-ax^2)cos(kx)dx=sqrt(pi/a)e^(-k^2/4a).
(20)

See also

Chi, Damped Exponential Cosine Integral, Nielsen's Spiral, Shi, Sine Integral

Related Wolfram sites

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

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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

Cosine Integral

Cite this as:

Weisstein, Eric W. "Cosine Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CosineIntegral.html

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