The sine and cosine integrals accumulate and , 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 , its small-argument expansion is . Integrating 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 , the sine integral approaches . Exponential damping gives , so the limiting constant follows by letting . Integration by parts of the tail gives .
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A trigonometric integral is a type of integral that involves trigonometric functions such as sine (sin), cosine (cos), tangent (tan), and their reciprocals or inverses. These integrals often arise in a variety of contexts, including physics, engineering, and mathematics, particularly in calculus when dealing with periodic functions or problems involving angles.