Fresnel’s sine and cosine integrals S(z) and C(z) are widely used in physics, particularly in optics and electromagnetic theory where they arise in the description of near-field Fresnel diffraction phenomena. The functions are defined in a variety of equivalent forms in the mathematical literature [29–30]. For example, the functions S(z) and C(z) are defined by the oscillatory integrals
\(S\left(z\right)={\int }_{0}^{z}\text{sin}\left(\frac{\pi }{2} {t}^{2}\right)dt\) (1),
\(C\left(z\right)={\int }_{0}^{z}\text{cos}\left(\frac{\pi }{2} {t}^{2}\right)dt\) (2).
The functions are also defined in the forms of the pairs S1(z), C1(z) and S2(z), C2(z)where
\({S}_{1}\left(z\right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{z}\text{sin}\left( {t}^{2}\right)dt\) (3),
\({C}_{1}\left(z\right)=\sqrt{\frac{2}{\pi }}{\int }_{0}^{z}\text{cos}\left( {t}^{2}\right)dt\) (4),
\({S}_{2}\left(z\right)=\frac{1}{\sqrt{2\pi }}{\int }_{0}^{z}\frac{\text{sin}\left( t\right) }{\sqrt{t}}dt=\frac{1}{2}{\int }_{0}^{z}{J}_{\frac{1}{2}}\left(t\right)dt\) (5),
\({C}_{2}\left(z\right)=\frac{1}{\sqrt{2\pi }}{\int }_{0}^{z}\frac{\text{cos}\left( t\right) }{\sqrt{t}}dt=\frac{1}{2}{\int }_{0}^{z}{J}_{-\frac{1}{2}}\left(t\right)dt\) (6),
where J1/2(t) and J− 1/2(t) refer to the ordinary Bessel functions of the first kind of orders ½ and -½, respectively. The three pairs of functions in (1,2), (3,4) and (5,6) are related to each other by [29]
\(S\left(z\right)={S}_{1}\left(\sqrt{\frac{\pi }{2}} z\right)= {S}_{2}\left(\frac{\pi }{2}{z}^{2}\right)\) (7),
\(C\left(z\right)={C}_{1}\left(\sqrt{\frac{\pi }{2}} z\right)= {C}_{2}\left(\frac{\pi }{2}{z}^{2}\right)\) (8).
From the computational point of view, all forms of the Fresnel sine and cosine integrals can be computed using the same algorithm with a slight change in the argument as explained in Eqs. (7,8).
Considering the form of the integrals given in (1,2), it is easy to show that they can be also expressed in terms of the Faddeyeva or “Faddeeva” function, \(\mathcal{w}\left(z\right)\) [31–35] where;
\(S\left(z\right)=\frac{-1-i}{4}\left[\left(1-{e}^{{\left(\frac{1-i}{2}\sqrt{\pi } z\right)}^{2}}\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } z\right)\right)+i\left(1-{e}^{{\left(\frac{1+i}{2}\sqrt{\pi } z\right)}^{2}}\mathcal{w}\left(\frac{1+i}{2}\sqrt{\pi } z\right)\right)\right]\) (9),
\(C\left(z\right)=\frac{-1+i}{4}\left[\left(1-{e}^{{\left(\frac{1-i}{2}\sqrt{\pi } z\right)}^{2}}\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } z\right)\right)-i\left(1-{e}^{{\left(\frac{1+i}{2}\sqrt{\pi z}\right)}^{2}}\mathcal{w}\left(\frac{1+i}{2}\sqrt{\pi } z\right)\right)\right]\) (10).
Using the expressions (9) and (10) with some mathematical manipulation, one can show that
$$S\left(x\right)=-\frac{1}{2}+\frac{1}{2}\mathfrak{ }\mathfrak{R}\left(\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } x\right)\right)\times \left(\text{cos}\left(\frac{\pi }{2}{x}^{2}\right)+\text{sin}\left(\frac{\pi }{2}{x}^{2}\right)\right)$$
\(+\frac{1}{2}\mathfrak{T}\left(\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } x\right)\right)\times \left(\text{sin}\left(\frac{\pi }{2}{x}^{2}\right)-\text{cos}\left(\frac{\pi }{2}{x}^{2}\right)\right)\) (11),
with\(S\left(iy\right)=-iS\left(y\right)\)
Similarly;
$$C\left(x\right)=-\frac{1}{2}+\frac{1}{2}\mathfrak{ }\mathfrak{R}\left(\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } x\right)\right)\times \left(\text{cos}\left(\frac{\pi }{2}{x}^{2}\right)-\text{sin}\left(\frac{\pi }{2}{x}^{2}\right)\right)$$
\(+\frac{1}{2}\mathfrak{T}\left(\mathcal{w}\left(\frac{1-i}{2}\sqrt{\pi } x\right)\right)\times \left(\text{cos}\left(\frac{\pi }{2}{x}^{2}\right)+\text{sin}\left(\frac{\pi }{2}{x}^{2}\right)\right)\) (12),
where\(C\left(iy\right)=iC\left(y\right)\)
The formulation of the functions as direct expressions in Faddeyeva function as in Eqs. (9,10) and in Eqs. (11,12) is not common in the literature although equivalent alternatives can be found.
Fresnel integrals are analytical functions defined for all complex values of z, over the whole complex plane. They are odd functions with the following symmetry relations
\(S\left(-z\right)=-S\left(z\right), S\left(\stackrel{-}{z}\right)= \stackrel{-}{S\left(z\right)}\) (13),
\(C\left(-z\right)=-C\left(z\right), C\left(\stackrel{-}{z}\right)= \stackrel{-}{C\left(z\right)}\) (14). In addition, the integrals have simple limiting values at z=zero and z→∞ where
\(S\left(0\right)=0, \underset{z\to \pm \infty }{\text{lim}}S\left(z\right)=\pm \frac{1}{2}\) , (15),
\(C\left(0\right)=0, \underset{z\to \pm \infty }{\text{lim}}C\left(z\right)=\pm \frac{1}{2}\) (16).
It is also worth mentioning that the Fresnel integrals S(z) and C(z) have the following simple converging series representations for small\(\left|z\right|.\)
$$S\left(z\right)={z}^{3}\sum _{k=0}^{\infty }\frac{{2}^{-2k-1}{\pi }^{2k+1}{\left(-{z}^{4}\right)}^{k}}{\left(4k+3\right)\left(2k+1\right)!}$$
(17),
$$C\left(z\right)=z\sum _{k=0}^{\infty }\frac{{2}^{-2k}{\pi }^{2k}{\left(-{z}^{4}\right)}^{k}}{\left(4k+1\right)\left(2k\right)!}$$
(18),
and the asymptotic expressions for z→∞ and |arg z|<π/2, in terms of the auxiliary functions or integrals \(f\left(z\right)\) and \(g\left(z\right)\) where
\(S\left(z\right)= \frac{1}{2}-f\left(z\right)\text{cos}\left(\frac{\pi }{2} {z}^{2}\right)-g\left(z\right)\text{sin}\left(\frac{\pi }{2} {z}^{2}\right)\) (19),
\(C\left(z\right)= \frac{1}{2}+f\left(z\right)\text{sin}\left(\frac{\pi }{2} {z}^{2}\right)-g\left(z\right)\text{cos}\left(\frac{\pi }{2} {z}^{2}\right)\) (20).
The auxiliary functions \(f\left(z\right)\) and \(g\left(z\right)\) are defined by the integrals
\(f\left(z\right)= \frac{1}{\pi \sqrt{2}} {\int }_{0}^{\infty }\frac{{e}^{-{z}^{2}t}}{\sqrt{t}\left({t}^{2}+1\right)} dt={\int }_{0}^{\infty }{e}^{-\pi zt}\text{cos}\left(\frac{\pi }{2}{t}^{2}\right)dt\) (21),
\(g\left(z\right)= \frac{1}{\pi \sqrt{2}} {\int }_{0}^{\infty }\frac{{\sqrt{t} e}^{-{z}^{2}t}}{\left({t}^{2}+1\right)} dt={\int }_{0}^{\infty }{e}^{-\pi zt}\text{sin}\left(\frac{\pi }{2}{t}^{2}\right)dt\) (22).