Question

From the results of the previous two problems, derive the relations $$ \begin{aligned} J_{-n-1 / 2}(z) &=\sqrt{\frac{2}{\pi}} z^{n+1 / 2}\left(\frac{1}{z} \frac{d}{d z}\right)^{n}\left(\frac{\cos z}{z}\right) \\ J_{n+1 / 2}(z) &=\sqrt{\frac{2}{\pi}} z^{n+1 / 2}\left(-\frac{1}{z} \frac{d}{d z}\right)^{n}\left(\frac{\sin z}{z}\right) \end{aligned} $$

Short answer

The derived formulas for \( J_{-n-1 / 2}(z) \) and \( J_{n+1 / 2}(z) \) are \( J_{-n-1 / 2}(z) = \sqrt{\frac{2}{\pi}} z^{\frac{n+1}{2}-2} (-\sin z - \frac{\cos z}{z}) \) and \( J_{n+1 / 2}(z) = \sqrt{\frac{2}{\pi}} z^{\frac{n+1}{2}-2} (\cos z - \frac{\sin z}{z}) \) respectively

Step-by-step solution

Differentiating the parts

Differentiate the \( \frac{\cos z}{z} \) and \( \frac{\sin z}{z} \) parts using the quotient rule. The quotient rule of differentiation states that the derivative of a quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Therefore, using the quotient rule, the derivative \( \frac{d}{dz} \left( \frac{\cos z}{z} \right) = \frac{z(-\sin z) - \cos z}{z^2} \) and the derivative \( \frac{d}{dz} \left( \frac{\sin z}{z} \right) = \frac{z \cos z - \sin z}{z^2} \)

Multiply by the power of z

According to the given formulas, the derivatives obtained in Step 1, needs to be multiplied with \( z^{n+1 / 2} \). Therefore, the step leads to \( z^{n+1 / 2} \left(\frac{z(-\sin z) - \cos z}{z^2}\right) \) for \( J_{-n-1 / 2}(z) \) and \( z^{n+1 / 2} \left(\frac{z \cos z - \sin z}{z^2}\right) \) for \( J_{n+1 / 2}(z) \) .

Simplifying the expressions

Simplify the expressions from the previous step. The simplification would result in \( z^{\frac{n+1}{2}-2} (-\sin z - \frac{\cos z}{z}) \) for \( J_{-n-1 / 2}(z) \) and \( z^{\frac{n+1}{2}-2} (\cos z - \frac{\sin z}{z}) \) for \( J_{n+1 / 2}(z) \)

Key concepts

Mathematical Physics

Bessel functions are a family of solutions to Bessel's differential equation that arise in various problems of wave propagation, static potentials, and when solving the Schrödinger equation in cylindrical or spherical coordinates. The study of these functions is significant in mathematical physics where they are used to describe the behaviors of waves and heat conduction in a cylindrical geometry. For instance, in quantum mechanics, Bessel functions can represent the radial part of the wave function for a particle confined to a cylindrical region.

The functions mentioned in the exercise, denoted as \( J_{-n-1/2}(z) \) and \( J_{n+1/2}(z) \), have a pattern related to half-integer indices, which hold great importance in solving boundary value problems in physics. The recurrence relations derived using the differential equations that define Bessel functions also simplify the computation of the functions for different orders, hence facilitating analytical work in physics problems.

Quotient Rule of Differentiation

The Quotient Rule is an important tool when tackling calculus problems involving derivatives of ratios of functions. It provides a method to differentiate expressions where one function is divided by another. Mathematically, if you have two differentiable functions \( u(z) \) and \( v(z) \), the derivative of their quotient \( \frac{u(z)}{v(z)} \) is given by:

\[ \frac{d}{dz}\left(\frac{u(z)}{v(z)}\right) = \frac{v(z)\cdot u'(z) - u(z)\cdot v'(z)}{\left(v(z)\right)^2} \]
As seen in the exercise, applying this rule can sometimes lead to a complex expression, and further simplification might be necessary. It's essential in mathematical physics where it is often required to differentiate complex ratios to find solutions to physical problems. Moreover, understanding and applying the quotient rule is a foundational skill for students aiming to excel in advanced calculus and physics.

Derivatives

Derivatives represent the rate of change of a function with respect to a variable. In physical terms, it's akin to finding the velocity of an object when given its position as a function of time. Derivatives are fundamental to calculus, with applications across science and engineering. The calculation done in the exercise is a practical demonstration of how to apply derivatives to Bessel functions.

Understanding derivatives is crucial while working with Bessel functions, especially when dealing with physical problems involving time or spatial variations, such as vibrations, tides, or heat distribution. These functions often require taking derivatives of various orders, as shown in the given exercise with the operator \( \left(-\frac{1}{z} \frac{d}{dz}\right)^n \), where \( n \) indicates the nth derivative with respect to \( z \).

Simplifying Mathematical Expressions

Simplifying mathematical expressions is not just about making an expression look 'neater'; it's a crucial step towards solving complex problems efficiently. It involves reducing fractions, eliminating complex denominators, combining like terms, and applying algebraic rules to achieve a more workable form of the expression. This skill is vital for analytical work in calculus and physics, where simplified expressions can reveal hidden relationships or simplify further operations, like integrating or taking higher derivatives.

In the context of Bessel functions from the exercise, after applying the quotient rule, additional steps were necessary to simplify the results. This entails canceling out common factors and consolidating the powers of \( z \) to reach the final expression for the Bessel functions. Mastering the art of simplification not only makes it easier to follow the subsequent mathematical procedures but also aids in gaining deeper insights into the underlying physical phenomena the equations represent.