sine and cosine integrals

The function (sin t)/t — the central shape of every diffraction pattern and antenna lobe — looks tame but has no elementary antiderivative. To work with the area under it we again name the integral. The sine integral Si(x) and cosine integral Ci(x) are the named running areas of (sin t)/t and (cos t)/t, the workhorses for oscillatory signals that decay only slowly.

Definitions: Si(x) = integral from 0 to x of (sin t)/t dt, which starts at 0, rises, and oscillates while converging to the limit Si(infinity) = pi/2. Cosine is trickier near zero because (cos t)/t blows up there, so we measure from infinity inward: Ci(x) = minus integral from x to infinity of (cos t)/t dt, which behaves like gamma_E plus ln x near the origin (gamma_E is the Euler-Mascheroni constant). Both are entire, smooth, oscillatory functions whose ripples shrink like 1/x; for large x they have the asymptotic forms Si(x) is approximately pi/2 minus (cos x)/x and Ci(x) is approximately (sin x)/x.

These integrals quantify the ringing and overshoot of band-limited signals. The Gibbs phenomenon — the persistent 9% overshoot near a jump in a truncated Fourier series — is expressed exactly through Si, whose first maximum Si(pi) is approximately 1.852 sets the overshoot height. Si and Ci also appear in antenna radiation resistance, optical diffraction, and the impulse response of ideal filters. They are the sinusoidal cousins of the exponential and logarithmic integrals, completing that family of named oscillatory antiderivatives.