Question

Use the derivative method to obtain as a second solution of Bessel's equation for the case when \(v=0\) the following expression: $$ J_{0}(z) \ln z-\sum_{n=1}^{\infty} \frac{(-1)^{n}}{(n !)^{2}}\left(\sum_{r=1}^{n} \frac{1}{r}\right)\left(\frac{z}{2}\right)^{2 n}, $$ given that the first solution is \(J_{0}(z)\) as specified by (16.63).

Short answer

The second solution is \( J_{0}(z) \ln z - \sum_{n=1}^\rightarrow \frac{(-1)^n}{(n!)^2} \sum_{r=1}^{n}\frac{1}{r}\bigg(\big(\frac{z}{2}\big)^{2n}\bigg) \

Step-by-step solution

- Recognize the Primary and Secondary Solutions

Bessel's equation of order zero has two solutions. The first solution is given as \( J_{0}(z)\). The task is to find the second solution using the derivative method.

- Define the Derivative Method

In this method, let's use the fact that if \( J_{v}(z) \) is a solution of Bessel's equation, another linearly independent solution can be found as \[ Y_{v}(z) = C J_{v}(z) \text{ln} z + \text{sum of series terms}. \]

- Apply the Derivative Expansion

For \( v = 0 \), the second solution can be written as \[ Y_{0}(z) = C J_{0}(z) \text{ln} z + \text{sum of series terms} \] where the series accounts for non-logarithmic components.

- Substitute into Bessel's Equation

Differentiate \( J_{0}(z) \) term by term to find \( J_{0}'(z) \). Use the logarithmic derivative technique and integrate resulting series.

- Obtain the Series Expression

Integrate the above series term by term to obtain: \[ J_{0}(z) \text{ln} z - \text{series sum} \] Summation over \( n \) and \( r \) to derive the necessary terms from integrations.

- Formulate the Solution

That completes the derivative method solution to find the unique second solution.

Key concepts

Derivative Method

The Derivative Method is a powerful tool for finding independent solutions to differential equations, including Bessel's equation. Bessel's equation of order zero can be written as:
\frac{d^2 y}{dz^2} + \frac{1}{z}\frac{dy}{dz} + \big(1 - \frac{u^2}{z^2}\big)y = 0 For the case when \(u = 0\), the first solution, \(J_0(z)\), is well-known. However, there exists a secondary solution involving a logarithmic term.
For Bessel’s equation, if \(J_ν(z)\) is a solution, a linearly independent solution, \( Y_ν(z)\), can be found using:
Y_ν(z) = C J_ν(z) \text{ln} z + \text{sum of series terms}. This method involves differentiating the known solution \(J_0(z)\) and integrating the terms step-by-step to form the second solution involving the logarithmic component and a series expansion.

Series Expansion

Series Expansion is integral in accurately expressing the second solution of Bessel's equation. After expressing the primary solution with a logarithmic component, we need to determine the remaining series terms without logarithms.
For \( v = 0 \), the second solution can be written as:
\[ Y_{0}(z) = C J_{0}(z) \text{ln} z + \text{sum of series terms} \]
To determine these series terms, use the method of differentiation and integration. Differentiate \( J_{0}(z) \) term by term, yielding expressions with terms involving \( \frac{z}{2}^{2n} \). Integrating each term results in a series sum:
\frac{(-1)^{n}}{(n!)^2}\bigg(\frac{z}{2}\bigg)^{2n}\bigg(\frac{1}{r}\bigg), which forms part of the secondary solution to Bessel's equation. By summing over all integral values of n and r, the series accurately reconstructs the solution.

Independent Solutions

Finding independent solutions to Bessel’s equation is essential for a complete set of solutions. Generally, differential equations of second order require two linearly independent solutions.

For Bessel's equation of order zero, \(J_{0}(z)\) is the primary solution. The second solution, found through the Derivative Method, includes logarithmic terms and compensating series expansions:

\(J_{0}(z) \text{ln} z - \text{sum of series terms}\).
The general form of the second solution thus combines logarithmic differentiation of \(J_{0}(z)\) with integrated series terms, producing a function orthogonal and independent of \(J_{0}(z)\). This ensures completeness in solving Bessel’s equation, crucial for practical applications and deeper mathematical insights.