Show that the Fresnel integrals are related to Bessel functions of half- integral order by $$ \begin{aligned} &C(u)=J_{1 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{5 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{9 / 2}\left(\frac{\pi}{2} u^{2}\right)+\cdots \\ &S(u)=J_{3 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{7 / 2}\left(\frac{\pi}{2} u^{2}\right)+J_{11 / 2}\left(\frac{\pi}{2} u^{2}\right)+\ldots \end{aligned} $$

Bessel functions are a series of solutions to Bessel's differential equation that appear in various fields of physics, including wave propagation and static potentials. Of particular interest are Bessel functions of half-integral order, which take the form \( J_{n+1/2}(x) \) where \( n \) is an integer. What makes these functions stand out is their connection to trigonometric functions, leading to simpler expressions.

When it comes to understanding Bessel functions of half-integral order, the connection with sine and cosine functions can be used to compute them more easily than Bessel functions of other orders. This property is especially useful in solving problems involving waveforms like the Fresnel integrals. By associating the Fresnel integrals with the appropriate Bessel functions, as demonstrated in the given exercise, we are translating the problem into the language of Bessel functions which can then be evaluated using their known properties.

Integral representation refers to expressing functions as integrals with respect to a certain variable. In the context of Bessel functions, the integral representation is a powerful tool because it offers a way to calculate these functions by integrating products of trigonometric functions over certain limits.

For example, the integral representation can turn the abstract concept of Bessel functions into a more tangible form involving integration of sine and cosine functions over an interval. This is illustrated in the step-by-step solution where the Fresnel integrals \( C(u) \) and \( S(u) \) are expressed in terms of double integrals involving cosine functions. Relating the complex Fresnel integrals to the more familiar trigonometric functions helps deepen our understanding and provides a clearer approach to computation.

The concept of series representation is crucial for functions like Bessel functions, as it provides a way to express these functions as an infinite sum of terms. Each term in the series representation of a Bessel function is given by a certain mathematical expression involving powers and factorials. This series form is particularly useful for approximating the function’s values and understanding its behavior.

In solving problems, as you move from the integral representation to the series representation, you gain a different insight into the function's qualities. As demonstrated in the Fresnel integrals exercise, upon transforming the integral representation to series form, we can see how \( C(u) \) and \( S(u) \) expand into sums involving Bessel functions of half-integral order. The series representation can often be easier to work with, especially in computational settings or when we're looking for approximate solutions.