Question

Bessel functions \(J_{p}(\lambda x)\) are solutions of \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(\lambda^{2} x^{2}-p^{2}\right) y=\) 0. Assume that \(x \in(0,1)\) and that \(J_{p}(\lambda)=0\) and \(J_{p}(0)\) is finite.s a. Show that this equation can be written in the form $$ \frac{d}{d x}\left(x \frac{d y}{d x}\right)+\left(\lambda^{2} x-\frac{p^{2}}{x}\right) y=0 $$ This is the standard Sturm-Liouville form for Bessel's equation. b. Prove that $$ \int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) d x=0, \quad \lambda \neq \mu $$ by considering $$ \int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] d x $$ Thus, the solutions corresponding to different eigenvalues \((\lambda, \mu)\) are orthogonal. c. Prove that $$ \int_{0}^{1} x\left[J_{p}(\lambda x)\right]^{2} d x=\frac{1}{2} J_{p+1}^{2}(\lambda)=\frac{1}{2} J_{p}^{\prime 2}(\lambda) $$

Short answer

First, we rewrote the Bessel equation into Sturm-Liouville Form. Then, using integrals, we proved the orthogonality of two Bessel functions for different eigenvalues. Finally, we proved a mathematical equality involving Bessel function using integration by parts and the formula for the derivative of the Bessel function of the first kind.

Step-by-step solution

Sturm-Liouville Form

First step is to rewrite the given Bessel function equation \(x^{2} y^{\prime \prime}+x y^{\prime}+\left(\lambda^{2} x^{2}-p^{2}\right) y=0\) to the Sturm-Liouville Form. This can be done by the operation \(\frac{d}{dx}(x \frac{dy}{dx})\) which allows to express the equation in the form \(\frac{d}{dx}\left(x\frac{dy}{dx}\right)+\left(\lambda^{2}x-\frac{p^{2}}{x}\right)y=0\).

Proving Orthogonality

The second part involves proving orthogonality of two Bessel functions. This requires integrating the given integrand and then simplifying using the Bessel's equation we have: \(\int_{0}^{1}\left[J_{p}(\mu x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\lambda x)\right)-J_{p}(\lambda x) \frac{d}{d x}\left(x \frac{d}{d x} J_{p}(\mu x)\right)\right] dx = 0\). This shows that when \(\lambda ≠ \mu\), \(\int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) dx = 0\) thus proving the orthogonality of the Bessel's functions for different eigenvalues.

Proof of the Equality involving Bessel Function

The final part requires proving a mathematical equality that involves the square of Bessel function. To prove \(\int_{0}^{1} x\left[J_{p}(\lambda x)\right]^{2} dx = \frac{1}{2} J_{p+1}^{2}(\lambda) = \frac{1}{2} J_{p}^{\prime 2}(\lambda)\) integration by parts is required. This will involve using the derivative of the Bessel function and the formula for the derivative of the Bessel function of the first kind. Implementation of these will lead us to prove the required equality.

Key concepts

Sturm-Liouville form

The concept of Sturm-Liouville form acts as a cornerstone in understanding various physical situations, particularly those involving differential equations. When we come across the Bessel function equation, \( x^{2} y'' + x y' + (\lambda^{2} x^{2} - p^{2}) y = 0 \), it can be transformed into the Sturm-Liouville form. This is more than a mere algebraic manipulation; it is a way of expressing differential equations so that they reveal more of their structure.

By performing the derivative operation \( \frac{d}{dx}(x \frac{dy}{dx}) \) and rearranging terms, we can bring the Bessel equation into the format \( \frac{d}{dx}\left(x\frac{dy}{dx}\right) + \left(\lambda^{2}x - \frac{p^{2}}{x}\right)y=0 \), which is the required Sturm-Liouville form. This form is particularly useful because it hints at certain properties of the solutions, like orthogonality and the existence of eigenvalues, which are essential for solving physical problems involving quantum mechanics, heat transfer, and vibrations.

Orthogonality of Bessel functions

Moving forward from the equation forms, we arrive at the Orthogonality of Bessel functions, a property that has far-reaching implications in mathematical physics. We say that two functions are orthogonal over an interval if their product integrated over that interval vanishes. In formal terms, for Bessel functions \( J_p(\lambda x) \) and \( J_p(\mu x) \), with distinct eigenvalues \( \lambda \) and \( \mu \), their orthogonality is expressed as \( \int_{0}^{1} x J_{p}(\lambda x) J_{p}(\mu x) dx = 0 \) when \( \lambda eq \mu \).

This orthogonality is crucial when dealing with problems, such as solving partial differential equations using separation of variables, where the solutions are expressed as a sum of orthogonal functions. It helps in simplifying the calculations and ensuring the uniqueness of solutions for physical problems.

Mathematical methods in physics

And finally, we delve into the Mathematical methods in physics, which are the toolkit that physicists use to model the natural world. Bessel functions are a prime example of these tools, often emerging in situations with cylindrical symmetry, like the vibrations of a circular drumhead or the radial electric field in a cylindrical capacitor.

The orthogonality property mentioned earlier is leveraged in methods such as Fourier series and eigenfunction expansions. These mathematical tools allow physicists to decompose complex problems into simpler, more manageable parts. By understanding the underlying mathematics, including the Sturm-Liouville theory and orthogonality of functions like Bessel's, students gain the ability to tackle challenging problems in quantum mechanics, electromagnetic theory, and beyond.

Overall, the fusion of these mathematical methods with physical insight leads to a deep understanding of the universe's workings, illustrating the harmonious interplay between mathematics and physics.