On the Fresnel Integrals and the Convolution

The Fresnel cosine integral C(x), the Fresnel sine integral S(x), and the associated functions C + (x), C − (x), S + (x), and S − (x) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of the Fresnel cosine integral and its associated functions with x r + and x r are evaluated. The Fresnel cosine integral C(x) is defined by C(x) = 2 π x 0 cos u 2 du, (1) (see [3]) and the associated functions C + (x) and C − (x) are defined by C + (x) = H(x)C(x), C − (x) = H(−x)C(x). (2) The Fresnel sine integral S(x) is defined by S(x) = 2 π x 0 sin u 2 du, (3) (see [3]) and the associated functions S + (x) and S − (x) are defined by S + (x) = H(x)S(x), S − (x) = H(−x)S(x), (4) where H denotes Heaviside's function. We define the function I r (x) by I r (x) = x 0 u r cos u 2 du (5) I 0 (x) = π 2 C(x), I 1 (x) = 1 2 sin x 2 , I 2 (x) = 1 2 x sin x 2 − √ π 2 √ 2 S(x). (6) We define the functions cos + x, cos − x, sin + x, and sin − x by cos + x = H(x) cos x, cos − x = H(−x) cos x, sin + x = H(x) sin x, sin − x = H(−x) sin x.