CI

Ci, the cosine integral, is a classical special function in mathematics and its applications. It is denoted Ci(x) and is defined for real x by an improper integral or by an equivalent integral representation. A common definition for x > 0 is Ci(x) = -∫_x^∞ (cos t)/t dt. An alternative form is Ci(x) = γ + ln x + ∫_0^x (cos t − 1)/t dt, where γ is the Euler–Mascheroni constant.

Properties and relations: Ci'(x) = cos x / x, linking the function to oscillatory behavior. Ci(x) is even more conveniently analyzed together with the sine integral Si(x) = ∫_0^x (sin t)/t dt, as both arise from integrating oscillatory kernels. Near zero, Ci(x) has a logarithmic singularity: Ci(x) ~ γ + ln x as x → 0+. As x → ∞, Ci(x) tends to zero with oscillations, and it admits an asymptotic expansion such as Ci(x) ~ sin x / x − cos x / x^2 + O(1/x^3).

Series and numerics: Ci(x) has a convergent power series about x = 0 given by Ci(x) = γ + ln x + ∑_{n=1}^∞ (−1)^n x^{2n} / [2n (2n)!]. This makes practical computation straightforward for small x, while the asymptotic form aids large-x evaluation.

Applications: Ci appears in the evaluation of integrals with oscillatory kernels, in diffraction theory, wave propagation, signal processing, and various Fourier transform problems. It also features in solutions to certain differential equations and in the study of asymptotic behavior of integrals.

Extensions: Ci is defined for complex arguments with a branch cut along the negative real axis, allowing analytic continuation beyond the real line.