Fresnel integral
The Fresnel integrals and are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations: formula_1 The simultaneous parametric plot of and is the Euler spiral (also known as the Cornu spiral or clothoid). Definition. The Fresnel integrals admit the following power series expansions that converge for all : formula_2 Some widely used tables use instead of for the argument of the integrals defining and . This changes their limits  at infinity from to and the arc length for the first spiral turn from to 2 (at ). These alternative functions are usually known as normalized Fresnel integrals. Euler spiral. The Euler spiral, also known as Cornu spiral or clothoid, is the curve generated by a parametric plot of against . The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering. From the definitions of Fresnel integrals, the infinitesimals and are thus: formula_3 Thus the length of the spiral measured from the origin can be expressed as formula_4 That is, the parameter is  the curve length measured from the origin , and the Euler spiral has infinite length. The vector also expresses the unit tangent vector along the spiral, giving . Since is the curve length, the curvature can be expressed as formula_5 Thus the rate of change of curvature with respect to the curve length is formula_6 An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: if a vehicle follows the spiral at unit speed , the parameter in the above derivatives also represents the time. Consequently, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration. Sections from Euler spirals are commonly incorporated into the shape of rollercoaster loops to make what are known as clothoid loops. Properties. Limits as approaches infinity. The integrals defining and cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as goes to infinity are known: formula_10 The limits of and as the argument tends to infinity can be found by  using several methods. One of them uses a contour integral of the function formula_11 around the boundary of the sector-shaped region in the complex plane formed by the positive -axis, the bisector of the first quadrant with , and a circular arc of radius centered at the origin. As goes to infinity, the integral along the circular arc tends to formula_12 where polar coordinates were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis tends to the half Gaussian integral formula_13 Note too that because the integrand is an entire function on  the complex plane, its integral along the whole contour is zero. Overall, we must have formula_14 where denotes the bisector of the first quadrant, as in the diagram. To evaluate the left hand side, parametrize the bisector as formula_15 where ranges from 0 to . Note that the square of this expression is just . Therefore, substitution gives the left hand side as formula_16 Using Euler's formula to take real and imaginary parts of gives this as formula_17 where we have written to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting formula_18 and  then equating real and imaginary parts produces the following system of two equations in the two unknowns and : formula_19 Solving this for and gives the desired result. Generalization. The integral formula_20 is a confluent hypergeometric function and also an incomplete gamma function formula_21 which reduces to Fresnel integrals if real or imaginary parts are taken: formula_22 The leading term in the asymptotic expansion is formula_23 and therefore formula_24 For , the imaginary part of this equation in particular is formula_25 with the left-hand side converging for and the right-hand side being its analytical extension to the whole plane less  where lie the poles of . The Kummer transformation of the confluent hypergeometric function is formula_26 with formula_27 Numerical approximation. For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions converge faster. Continued fraction methods may also be used. For computation to particular target precision, other approximations have been developed. Cody developed a set of efficient approximations based on rational functions that give relative errors down to . A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder. Boersma  developed an approximation with error less than . Applications. The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects. More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve. Other applications are rollercoasters or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.