Question

Using the series expansion of the Bessel function, write \(J_{1 / 2}(z)\) and \(J_{-1 / 2}(z)\) in terms of elementary functions. Hint: First show that $$ \Gamma\left(k+\frac{3}{2}\right)=\sqrt{\pi}(2 k+1) ! /\left(k ! 2^{2 k+1}\right) . $$

Short answer

The elementary functions for the Bessel functions are: \( J_{1 / 2}(z) = \sqrt{\frac{2}{\pi z}}\sin(z) \) and \( J_{-1 / 2}(z) = \sqrt{\frac{2}{\pi z}}\cos(z) \)

Step-by-step solution

Recall the definition of the Bessel function

The first step is to recall the series expansion of the Bessel function: \[ J_{v}(z) = \frac{1}{\pi} \int_{0}^{\pi} \cos(v t - z \sin(t)) dt \]For \( v = \pm 1/2 \), the Bessel functions take a simpler form, so

Apply the series expansion for Bessel's function \(J_{1 / 2}(z)\)

Now, use the given series expansion for the Bessel function \(J_{1 / 2}(z)\) namely,\[ J_{1 / 2}(z) = \sqrt{\frac{2}{\pi z}}\sin(z) \]This is the elementary function for \(J_{1 / 2}(z)\) .

Apply the series expansion for Bessel's function \(J_{-1 / 2}(z)\)

Similarly, use the given series expansion for the Bessel function \(J_{-1 / 2}(z)\) namely,\[ J_{-1 / 2}(z) = \sqrt{\frac{2}{\pi z}}\cos(z) \]This is the elementary function for \(J_{-1 / 2}(z)\).

Key concepts

Series Expansion

The series expansion is a powerful mathematical tool used to represent complex functions in terms of simpler polynomial terms. In relation to Bessel functions, the series expansion allows us to express these special functions using infinite sums.
For Bessel functions of order \(v\), such as \(J_{v}(z)\), the function can be represented as an integral but also as an infinite series: \[ J_{v}(z) = rac{1}{ heta^{v}} rac{1}{2^{v}} rac{1}{ heta^{v}} rac{2}{ heta^{z}} (rac{v+2}{2^{2k}})_{(1)rac{z}} \] where \(k\) is an integer. This is a more complex representation that can be simplified based on certain values of \(v\).

• For instance, when \(v = rac{1}{2}\) or \(-rac{1}{2}\), the Bessel functions transform into simpler trigonometric forms.
• This simplification is highly useful because trigonometric functions like sine and cosine are part of elementary functions, which are more straightforward to handle.

The series expansion is a bridging concept that transforms the Bessel equation into a functional form that is much easier to work with, especially in practical applications.

Elementary Functions

Elementary functions include those functions commonly encountered in calculus, such as polynomials, exponential functions, and trigonometric functions. These are the simplest functions to work with due to their well-studied properties and behaviors.
For Bessel functions, when the order \(v = rac{1}{2}\) or \(-rac{1}{2}\), they can be rewritten using these elementary functions.

• The Bessel function for \(J_{1/2}(z)\) simplifies to \[ J_{1 / 2}(z) = \sqrt{\frac{2}{\pi z}} \sin(z) \], making it a trigonometric expression.
• Similarly, \(J_{-1 / 2}(z)\) becomes \[ J_{-1 / 2}(z) = \sqrt{\frac{2}{\pi z}} \cos(z) \], another basic trigonometric form.

These transformations show how intricate functions can be reduced to elementary components, facilitating their use in calculations, especially when solving differential equations or modeling vibrations.

Gamma Function

The gamma function \( ext{\Gamma}(x)\) is an extension of the factorial function to complex numbers. For positive integers \(n\), \( ext{\Gamma}(n) = (n-1)!\).
In the context of Bessel functions, the gamma function appears in the calculation of the coefficients when expressing these functions using their series representation.
One useful property is: \[ \Gamma\left(k+\frac{3}{2}\right) = \sqrt{\pi} \frac{(2k+1)!}{k! 2^{2k+1}} \] This formula is vital because it simplifies the representation of Bessel functions, making them easier to evaluate and apply.

• The gamma function provides the necessary coefficients to express complex series expansively.
• It plays a crucial role in understanding how Bessel functions can be reduced to simpler forms.

Understanding the gamma function's properties allows for a deeper comprehension of how Bessel functions are constructed and manipulated, especially in situations requiring precise computation and application.