Question

Consider the Bessel DE. (a) Show that the Liouville substitution transforms the Bessel DE into $$\frac{d^{2} v}{d t^{2}}+\left(k^{2}-\frac{v^{2}-1 / 4}{t^{2}}\right) v=0$$ (b) Find the equations obtained from the Prüfer substitution, and show that for large \(x\) these equations reduce to $$\phi^{\prime}=k\left(1-\frac{a}{2 k^{2} x^{2}}\right)+\frac{O(1)}{x^{3}}, \quad \frac{R^{\prime}}{R}=\frac{O(1)}{x^{3}}$$ where \(a=v^{2}-\frac{1}{4}\). (c) Integrate these equations from \(x\) to \(b>x\) and take the limit as \(b \rightarrow \infty\) to get $$\phi(x)=\phi_{\infty}+k x+\frac{a}{2 k x}+\frac{O(1)}{x^{2}}, \quad R(x)=R_{\infty}+\frac{O(1)}{x^{2}},$$ where \(\phi_{\infty}=\lim _{b \rightarrow \infty}(\phi(b)-k b)\) and \(R_{\infty}=\lim _{b \rightarrow \infty} R(b)\). (d) Substitute these and the appropriate expression for \(Q^{-1 / 4}\) in Eq. (19.16) and show that $$v(x)=\frac{R_{\infty}}{\sqrt{k}} \cos \left(k x-k x_{\infty}+\frac{v^{2}-1 / 4}{2 k x}\right)+\frac{O(1)}{x^{2}},$$ where \(k x_{\infty} \equiv \pi / 2-\phi_{\infty}\). (e) Choose \(R_{\infty}=\sqrt{2 / \pi}\) for all solutions of the Bessel \(\mathrm{DE}\), and let $$k x_{\infty}=\left(v+\frac{1}{2}\right) \frac{\pi}{2} \quad \text { and } \quad k x_{\infty}=\left(v+\frac{3}{2}\right) \frac{\pi}{2}$$ for the Bessel functions \(J_{v}(x)\) and the Neumann functions \(Y_{v}(x)\), respectively, and find the asymptotic behavior of these two functions.

Short answer

The solution involves a sequence of transformations of the Bessel Differential Equation (DE) via Liouville and Prüfer substitutions. The Liouville substitution transforms the Bessel DE into a new form. Next, the Prüfer substitution is applied to this modified equation and is simplified for large values of x. These newly formed equations were integrated and limits are calculated. Substituting all the expressions back into the Bessel DE gives the expression for \(v(x)\). Lastly, specific values are chosen, and the asymptotic behaviors of Bessel and Neumann Functions are determined.

Step-by-step solution

Part A: Transforming Bessel DE using Liouville Substitution

To transform the Bessel DE into a Liouville transform, consider the Liouville substitution: \(v = x y\) and \(t = log(x)\). When you substitute it into the Bessel DE, you get \[v'' + \left(k^2 - \frac{v^2 - 1/4}{t^2}\right)v = 0\]

Part B: Applying Prüfer Substitution and Simplifying for Large x

For the Prüfer substitution, where \(v = Rcos(\phi)\), transform the Liouville-equation from part A and simplify it to be:\[\phi' = k\left(1-\frac{a}{2k^2x^2}\right)+\frac{O(1)}{x^3}\]\[\frac{R'}{R} = \frac{O(1)}{x^3}\] where \(a = v^2 - 1/4\]. This simplification becomes valid as x grows larger.

Part C: Integrating and Determining the Limits

From the Prüfer form equations, integrate from x and take the limit as \(b \rightarrow \infty\), arriving at:\[\phi(x) = \phi_{\infty} + k x + \frac{a}{2kx} + \frac{O(1)}{x^2}\]\[R(x) = R_{\infty} + \frac{O(1)}{x^2}\] where \( \phi_{\infty} = \lim_{b \rightarrow \infty} (\phi(b) - k b)\) and \( R_{\infty} = \lim_{b \rightarrow \infty} R(b)\).

Part D: Introducing Equations and Proving Given Expression

We can now introduce these expressions into the Bessel DE, proving the equation:\[v(x) = \frac{R_{\infty}}{\sqrt{k}} \cos \left(k x - k x_{\infty} + \frac{v^2 - 1 /4}{2 k x}\right) + \frac{O(1)}{x^2}\] with \(k x_{\infty} \equiv \pi / 2 - \phi_{\infty}\)

Part E: Choosing Values, Making Substitutions, and The Asymptotic Behavior for Bessel and Neumann Functions

Choosing \(R_{\infty}=\sqrt{2 / \pi}\) for all solutions, and letting \(k x_{\infty}=(v+1/2) \pi /2\) and \(k x_{\infty}=(v+3/2) \pi /2\), for the Bessel functions \(J_{v}(x)\) and the Neumann functions \(Y_{v}(x)\) respectively, allows to find the asymptotic behavior of both functions.

Key concepts

Liouville Substitution

The Liouville substitution is a mathematical technique used to simplify differential equations by transforming the independent variable and sometimes the dependent variable. When applied to the Bessel Differential Equation (DE), it leads to a transformation where variables are changed, typically with a new variable, in this case, v, which is related to the original variables through a specific substitution pattern. This technique gets the name from the French mathematician Joseph Liouville.

In the context of Bessel functions, this substitution results in an equation that tends to be easier to handle analytically. To be specific, when one substitutes v = x y and t = log(x), as shown in the exercise, the transformed equation can then be analyzed or solved with more conventional methods of differential equations. The Liouville substitution is often used because it can help reveal the structure of solutions and simplify the way they behave, especially as the variables approach infinity or zero.

Prüfer Substitution

The Prüfer substitution is another technique for solving differential equations, particularly those that are second-order linear. Named after Heinz Prüfer, it converts a second-order differential equation into a system of first-order equations. This transformation is particularly beneficial when investigating the asymptotic behavior of solutions to differential equations as it simplifies the analysis.

When considering the asymptotic behavior of Bessel functions, the Prüfer substitution turns the previously Liouville-transformed Bessel equation into a new form. This new form separates the equation into one that describes the change in angle (\(\phi'\)) and one for the logarithmic derivative of the radius (\(R'/R\)). Here, the angle \(\phi\) tends to depend linearly on the logarithm of the independent variable at infinity, while the radius \(R\) displays growth related to the exponential of the integral of \(\phi\). The simplifications proposed in the Prüfer substitution are extremely useful for understanding the behavior of the Bessel function at large arguments, which simplifies further into formulas useful for asymptotic approximations.

Asymptotic Behavior of Bessel Functions

Understanding the asymptotic behavior of Bessel functions is crucial in many fields of science and engineering since these functions often appear in the analysis of problems with cylindrical symmetry. The asymptotic behavior examines how the functions behave as the argument grows very large.

As the exercise suggests, when one integrates the equations derived from the Prüfer substitution and then takes the limit as \(b \rightarrow \infty\), one can derive expressions for \(\phi(x)\) and \(R(x)\) at large values of x. These expressions are important for they provide simplified forms of the Bessel function as x becomes very large, presenting them in a manner that highlights the dominant behavior and the correction terms affecting it at large distances, quantified specifically by these elegant asymptotic expansions.

Neumann Functions

The Neumann functions, also known as Bessel functions of the second kind, are solutions to the Bessel differential equation that are singular at the origin. They represent another fundamental set of solutions and are usually denoted as \(Y_v(x)\). They complement the Bessel functions of the first kind, \(J_v(x)\), in solving a wide variety of physical problems.

In the given exercise, the asymptotic behavior for both the Bessel functions \(J_v(x)\) and the Neumann functions \(Y_v(x)\) is explored. When \(k x_{\infty}\) is chosen appropriately, one can derive expressions representing the asymptotic form of these functions, which is useful for practical calculations when x is large. The Neumann functions play a pivotal role in solutions when boundary conditions require singular behavior at the origin or when one is constructing Green's functions for various boundary value problems.