Establish the recursion relations \((2.3 .10)\) and \((2.3 .11)\) from the series \((2.3 .9)\) or directly from the differential equation (2.3.7). Note the special case \(d J_{0}(u) / d u=-J_{1}(u)\). Can you develop a recursion relation for the second derivative, \(d^{2} J_{m}(u) / d u^{2} ?\)

In the realm of mathematics, particularly in the study of Bessel functions, recursion relations serve as vital tools for constructing solutions to complex problems. These relations exploit the interconnection between Bessel functions of different orders to simplify calculations and solve differential equations more efficiently.

For instance, the recursion relation enables us to express the derivative of a Bessel function of a particular order, say, \(J_m(u)\), in terms of Bessel functions of neighboring orders. The first recursion relation that arises from the differentiation of the power series is \(\frac{dJ_m(u)}{du} = \frac{1}{2}(J_{m-1}(u) - J_{m+1}(u))\), which proposes a symmetrical relationship between the derivatives of the adjacent orders of Bessel functions.

Furthermore, recursion relations are particularly useful for generating series expansions for Bessel functions, computational purposes, and even in theoretical analysis where the behavior of these functions at various limits is of interest. Importantly, the understanding and application of recursion relations can immensely benefit students and researchers working with Bessel functions, whether in physical applications or in advancing mathematical theory.

Bessel functions emerge as solutions to a specific type of differential equation known as the Bessel differential equation. Taking the form \(u^2 \frac{d^2}{du^2} + u \frac{d}{du} + (u^2 - m^2))J_m(u) = 0\), this equation is of great significance in physics, engineering, and applied mathematics – primarily because it appears in problems that exhibit cylindrical symmetry, such as the vibrations of a circular drumhead or heat conduction in a cylindrical object.

Understanding the nature of differential equations is crucial, as they enable us to describe a wide range of phenomena. In this specific context, the Bessel differential equation is a second-order linear homogeneous differential equation. Such equations often feature variable coefficients (here, dependent on \(u\)) and their solutions, which are the Bessel functions, are uniquely determined by their order \(m\) and argument \(u\).

The relationship between the Bessel functions of different orders and their derivatives, through recursion relations, exemplifies the interconnected nature of the solutions to the Bessel differential equation. It also highlights how differential equations are not just abstract mathematical constructs, but tools that directly correspond to physical reality.

A cornerstone in mathematical analysis and its applications to solving differential equations is the power series expansion. The power series allows us to represent functions as sums of infinitely many terms, each of which is a power of a certain variable, scaled by a coefficient. For Bessel functions, the series expansion takes the form \(J_m(u) = \sum_{n=0}^\infty \frac{(-1)^n}{n! (n+m)!} \left(\frac{u}{2}\right)^{2n+m}\) where \(m\) denotes the order of the Bessel function, and \(u\) is the variable.

Expansions like these are exceptionally powerful: they can converge to the function over a certain interval, and they provide a practical means for calculating function values numerically. Additionally, they lay the groundwork for more complex operations – like differentiation and integration – which can often be performed term-wise. For Bessel functions, power series expansions are essential for deriving properties such as the aforementioned recursion relations, and for analyzing the behavior of these functions near the origin, where \(u\) is small.

By grasping the concept of power series expansion, students gain access to a versatile tool for approximating functions and for use in perturbation methods, perturbative expansions in quantum mechanics, and even in calculating probabilities in statistics, showcasing the far-reaching implications of this mathematical technique.