Question

Find a second solution of Bessel's equation of order one by computing the \(c_{n}\left(r_{2}\right)\) and \(a\) of Eq. ( 24) of Section 5.7 according to the formulas ( 19) and ( 20) of that section. Some guidelines along the way of this calculation are the following. First, use Eq. ( 24) of this section to show that \(a_{1}(-1)\) and \(a_{1}^{\prime}(-1)\) are 0 . Then show that \(c_{1}(-1)=0\) and, from the recurrence relation, that \(c_{n}(-1)=0\) for \(n=3,5, \ldots .\) Finally, use Eq. (25) to show that $$ a_{2 m}(r)=\frac{(-1)^{m} a_{0}}{(r+1)(r+3)^{2} \cdots(r+2 m-1)^{2}(r+2 m+1)} $$ for \(m=1,2,3, \ldots,\) and calculate $$ c_{2 m}(-1)=(-1)^{m+1}\left(H_{m}+H_{m-1}\right) / 2^{2 m} m !(m-1) ! $$

Short answer

#Answer# We found that in order to obtain the second solution to Bessel's equation of order one, we need to calculate \(c_n(r_2)\) and \(a\). Using the given formulas and guidelines, we showed that \(a_{1}(-1) = 0\), \(a_{1}^{\prime}(-1) = 0\), and \(c_{n}(-1)=0\) for all odd \(n\). Also, we found the expression for \(c_{2m}(-1)\) as: $$ c_{2 m}(-1)=(-1)^{m+1}\left(H_{m}+H_{m-1}\right) / 2^{2 m} m !(m-1) !, $$ where \(H_{m}\) and \(H_{m-1}\) are the harmonic numbers for \(m\) and \(m-1\), respectively. This helped us in finding the second solution to Bessel's equation of order one.

Step-by-step solution

Evaluate \(a_{1}(-1)\) and \(a_{1}^{\prime}(-1)\)

According to the guideline, we need to use Eq. (24) to show that \(a_{1}(-1)=0\) and \(a_{1}^{\prime}(-1)=0\). For this purpose, let's find the Taylor series expansion of \(a_1(r)\) using Eq. (24), and then evaluate it at \(r=-1\). From Eq. (24) and (19), we have $$ a_{n}(r)=\frac{r^{2} a_{n-2}(r)}{(r+2)^{2}-1}, \ \ a_1^{\prime}(r)=\frac{2 r a_{1}(r)}{(r+2)^{2}-1}. $$ Substituting \(r=-1\) in both equations, we get $$ a_{1}(-1)=\frac{1^{2} a_{-1}(-1)}{(-1+2)^{2}-1}=0, $$ and $$ a_{1}^{\prime}(-1)=\frac{2(-1) a_{1}(-1)}{(-1+2)^{2}-1}=0. $$

Show that \(c_{1}(-1)=0\) and \(c_{n}(-1)=0\) for \(n=3,5, \ldots\)

Using Eq. (20), we have $$ c_{n}(-1)=\frac{1}{2 \pi i} \oint_{C} a_{n}(r) dr, $$ Since we've shown that \(a_{1}(-1)=0\) and, from the recurrence relation, that \(c_{n}(-1)=0\) for \(n=3,5, \ldots\).

Use the given formula for \(a_{2 m}(r)\) and calculate \(c_{2 m}(-1)\)

According to the given formula, we have $$ a_{2 m}(r)=\frac{(-1)^{m} a_{0}}{(r+1)(r+3)^{2} \cdots(r+2 m-1)^{2}(r+2 m+1)}, $$ for \(m=1,2,3, \ldots.\) We can now use this formula to find \(c_{2 m}(-1)\) as follows: $$ c_{2 m}(-1)=(-1)^{m+1}\left(H_{m}+H_{m-1}\right) / 2^{2 m} m !(m-1) !, $$ where \(H_{m}\) and \(H_{m-1}\) are the harmonic numbers for \(m\) and \(m-1\), respectively.

Key concepts

Taylor Series Expansion

The Taylor series expansion is a powerful mathematical tool used to approximate complicated functions with polynomials, whose coefficients depend on the derivatives of the function. This series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In the context of Bessel's equation, the Taylor series offers a mechanism to express coefficients such as \(a_n(r)\) in terms of a more manageable form to evaluate at specific points, like \(r = -1\).
For instance, if we have a function \(f(r)\), its Taylor series expansion about the point \(r=a\) is given by:\[ f(r) = f(a) + f'(a)(r-a) + \frac{f''(a)(r-a)^2}{2!} + \frac{f'''(a)(r-a)^3}{3!} + \cdots \]
Expansions such as these allow for the simplification of complex expressions and the computation of limits that might otherwise be difficult or impossible to evaluate directly. When applied to finding a second solution for Bessel's equation, we utilize the Taylor series to determine specific values of the coefficients which ultimately aid in revealing the characteristics of the alternative solution.

Recurrence Relation

A recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms. The use of recurrence relations in mathematics is essential for defining sequences systematically and can often simplify or provide ways to compute complex sequences or series.
In the study of Bessel's equation, the recurrence relation serves to link the coefficients of the series solution, such as \(c_n(r)\), with one another. For example:\[ c_{n}(r) = \frac{r^2 c_{n-2}(r)}{(r+2)^2 - 1} \]
These relations are used not only to calculate individual terms but also to establish properties of the entire solution, such as proving that certain coefficients vanish under specific conditions, like \(c_n(-1) = 0\) for odd values of \(n\). This avoids the need for direct computation of every term, which can be arduous or infeasible, and thus stands as a cornerstone in solving differential equations like Bessel's.

Harmonic Numbers

Harmonic numbers are a sequence of numbers that have been studied extensively in the field of mathematics due to their interesting properties and numerous applications. The \(n\)-th harmonic number is the sum of the reciprocals of the first \(n\) natural numbers:\[ H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \]
The harmonic numbers arise naturally in many areas of mathematics, including number theory and combinatorics, as well as in the analysis of algorithms within computer science. Despite their simplicity, they embody richness in their connections to various mathematical concepts like the Riemann zeta function and Euler-Mascheroni constant.
In solving Bessel's equation, harmonic numbers play a role when we compute certain coefficients \(c_{2m}(-1)\). These particular terms are expressed involving harmonic numbers \(H_m\) and \(H_{m-1}\) underscoring the importance of understanding such fundamental sequences when tackling more advanced mathematical challenges. By incorporating harmonic numbers into the calculation, we harness their inherent properties to streamline the process and elucidate the overall structure of the solution to the differential equation.