Integral representation

An integral representation expresses a function through an integral of specified data and an integral kernel. Green-function representations solve differential equations this way, while the Bromwich inversion formula reconstructs a function from its Laplace transform.

The sine and cosine integrals accumulate $\sin t/t$ and $\cos t/t$, with different normalizations at zero and infinity. They arise in oscillatory integrals, Fourier transforms of singular kernels and Fourier series of logarithmic profiles.

For real $x>0$, its small-argument expansion is $\mathrm{Ci}(x)={\gamma }_E+\log x+O(x^2)$. Integrating ${\mathrm{Ci}}^{\prime }(x)=\cos x/x$ separates the logarithm from the regular correction. Its conventional constant is the Euler--Mascheroni constant; the primary series normalization is dlmf.nist.gov/6.6.E6 .

For real $x>0$, the sine integral approaches $\pi /2$. Exponential damping gives ${\int }_0^{\infty }{e}^{-at}\sin t/tdt=\arctan (1/a)$, so the limiting constant follows by letting $a\downarrow 0$. Integration by parts of the tail gives $\mathrm{Si}(x)=\pi /2-\cos x/x-\sin x/x^2+O(x^{-3})$.