Problem 10 - Differential Equations and Linear Algebra

Question

Given that $$ J_{1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \sin x, \quad J_{-1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \cos x $$ express \(J_{3 / 2}(x)\) and \(J_{-3 / 2}(x)\) in closed form. Convince yourself that all half-integer order Bessel functions of the first kind can be expressed as a finite sum of terms involving products of \(\sin x, \cos x,\) and powers of \(x\).

Step-by-Step Solution

Answer

In summary, we can express \(J_{3/2}(x)\) and \(J_{-3/2}(x)\) in closed form as follows: - \(J_{3/2}(x) = \sqrt{\frac{2}{\pi x}} (\sin{x} - \cos{x})\) - \(J_{-3/2}(x) = \sqrt{\frac{2}{\pi x}} (\cos{x} - \sin{x})\) These expressions, along with the half-integer Bessel functions of the first kind, can be derived using recursion formulas and trigonometric functions. This demonstrates that all half-integer order Bessel functions of the first kind can be expressed as a finite sum of terms involving products of sin(x), cos(x), and powers of x.

Step 1: Recursion Formulas for Bessel Functions

: Recall that Bessel functions of the first kind satisfy the recursion formulas: 1. \(J_{v-1}(x) + J_{v+1}(x) = \frac{2v}{x} J_v(x)\) 2. \(J_{v-1}(x) - J_{v+1}(x) = 2J_v'(x)\)

Step 2: Compute J_{3/2}(x) in terms of J_{1/2}(x) and J_{-1/2}(x)

: Using the first recursion formula, we have: \(J_{1/2}(x) = J_{3/2}(x) + J_{-1/2}(x)\). Now, using the given expressions for \(J_{1/2}(x)\) and \(J_{-1/2}(x)\), we get: \(J_{3/2}(x) = J_{1/2}(x) - J_{-1/2}(x)\) \(J_{3/2}(x) = \sqrt{\frac{2}{\pi x}} \sin{x} - \sqrt{\frac{2}{\pi x}} \cos{x}\)

Step 3: Compute J_{-3/2}(x) in terms of J_{1/2}(x) and J_{-1/2}(x)

: Using the first recursion formula again, we have: \(J_{-1/2}(x) = J_{-3/2}(x) + J_{1/2}(x)\). Now, using the given expressions for \(J_{1/2}(x)\) and \(J_{-1/2}(x)\), we get: \(J_{-3/2}(x) = J_{-1/2}(x) - J_{1/2}(x)\) \(J_{-3/2}(x) = \sqrt{\frac{2}{\pi x}} \cos{x} - \sqrt{\frac{2}{\pi x}} \sin{x}\)

Step 4: Consider Half-Integer Order Bessel Functions

: Given that Bessel functions of the first kind satisfy certain recursion formulas, we can observe that half-integer order Bessel functions can be expressed as a finite sum of terms involving products of sin(x), cos(x), and powers of x. This is demonstrated in the closed-form expressions for the Bessel functions of order 1/2, -1/2, 3/2, and -3/2. For example, we have: \(J_{3/2}(x) = \sqrt{\frac{2}{\pi x}} \sin{x} - \sqrt{\frac{2}{\pi x}} \cos{x}\) This observation is based on the recursion formulas and the simple trigonometric functions in the given expressions, which allow us to continue finding higher half-integer order Bessel functions as finite sums of trigonometric functions and powers of x.

Key Concepts

Half-Integer Order

Half-integer order Bessel functions are a special category of Bessel functions of the first kind. These functions are particularly interesting because they involve half-integer indices, like \( \frac{1}{2}, \frac{3}{2}, \frac{5}{2} \), and so on.

Distinctively, half-integer Bessel functions have solutions that can be expressed using trigonometric functions, which distinguishes them from those with integer orders. For example, the given Bessel functions for orders \( \frac{1}{2} \) and \(-\frac{1}{2} \) can be represented using sine and cosine functions respectively:

\[ J_{1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \sin x, \quad J_{-1 / 2}(x)=\sqrt{\frac{2}{\pi x}} \cos x \]

These expressions show that the half-integer order Bessel functions involve simple mathematical operations, such as multiplication with \( x \) in the denominator and the use of basic trigonometric functions.

Recursion Formulas

Bessel functions obey certain recursion relations that are crucial for deriving expressions for new orders based on known ones. These formulas relate Bessel functions of different orders.

The two main recursion formulas are:

• \( J_{v-1}(x) + J_{v+1}(x) = \frac{2v}{x} J_v(x) \)
• \( J_{v-1}(x) - J_{v+1}(x) = 2J_v'(x) \)

Using these formulas, you can compute Bessel functions of new orders by relating them to other known Bessel functions of half-integer orders.

For example, to find \( J_{3/2}(x) \), you use the first recursion formula:\[ J_{1/2}(x) = J_{3/2}(x) + J_{-1/2}(x) \]From this, you can easily isolate and compute \( J_{3/2}(x) \) by utilizing the known values of \( J_{1/2}(x) \) and \( J_{-1/2}(x) \).

• \( J_{3 / 2}(x) = \sqrt{\frac{2}{\pi x}} \sin x - \sqrt{\frac{2}{\pi x}} \cos x \)

This highlights how recursion formulas are essential tools in systematically building functions of higher orders.

Trigonometric Representations

One of the elegant features of half-integer order Bessel functions is their approximation through simple trigonometric functions. These representations primarily use the sine and cosine functions multiplied by specific powers and constants.

Observe how the expressions for the Bessel functions provided were written:

• For \( J_{\frac{1}{2}}(x) \), it was expressed as \( \sqrt{\frac{2}{\pi x}} \sin x \)
• For \( J_{- \frac{1}{2}}(x) \), it was represented as \( \sqrt{\frac{2}{\pi x}} \cos x \)

These trigonometric representations are not only mathematically pleasant but also efficient, permitting easy computation and simplification of higher order Bessel functions through series expansions or transformations.

This pattern allows us to understand and anticipate the formulation of further half-integer order Bessel functions since they follow a recognizable and stable structure. This is how the functions \( J_{\pm \frac{3}{2}}(x) \) were deduced from their given \( \pm \frac{1}{2} \) counterparts, showing that both cosine and sine are elemental in these representations.