Question

The Bessel equation of order zero is $$ x^{2} y^{\prime \prime}+x y^{\prime}+x^{2} y=0 $$ Show that \(x=0\) is a regular singular point; that the roots of the indicial equation are \(r_{1}=r_{2}=0 ;\) and that one solution for \(x>0\) is $$ J_{0}(x)=1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}} $$ Show that the series converges for all \(x .\) The function \(J_{0}\) is known as the Bessel function of the first kind of order zero.

Short answer

Answer: The Bessel function of the first kind of order zero, J_0(x), is defined as: $$ J_{0}(x)=1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}} $$

Step-by-step solution

Regular Singular Point

To confirm that \(x=0\) is a regular singular point, we need to check two conditions: 1. The coefficients of the equation are analytic at \(x=0\). 2. The point \(x=0\) is not an ordinary point. For the Bessel equation, the coefficients are \(x^2\), \(x\), and \(x^2\), which are all analytic at \(x=0\). Additionally, since the coefficients of \(y^\prime\) and \(y^{\prime \prime}\) have \(x\) in their denominator, these points are not ordinary points. Therefore, \(x=0\) is a regular singular point.

Roots of the Indicial Equation

Now, we'll find the roots of the indicial equation. Create a series solution of the form: $$ y(x) = \sum_{n=0}^\infty a_nx^{n+r} $$ Taking the derivatives of \(y(x)\): $$ y'(x) = \sum_{n=0}^\infty (n+r)a_nx^{n+r-1} $$ $$ y''(x) = \sum_{n=0}^\infty (n+r)(n+r-1)a_nx^{n+r-2} $$ Substitute the derivatives into the Bessel equation: $$ x^2y''(x) + xy'(x) + x^2y(x) = 0 \qquad (1) $$ Plug in the series representations of \(y(x)\), \(y'(x)\), and \(y''(x)\) into equation (1): $$ \sum_{n=0}^\infty [(n+r)(n+r-1)a_nx^{n+r} + (n+r)a_nx^{n+r} + a_nx^{n+r}] = 0 $$ Since this equation must hold for all \(x\), the terms inside the summation with \(x^{n+r}\) must all sum to zero: $$ (n+r)(n+r-1)a_n + (n+r)a_n + a_n = 0 $$ For \(n = 0\), the terms simplify to \(r^2a_0 + ra_0 + a_0 = 0\). Factoring out \(a_0\), we have \((r^2 + r + 1)a_0 = 0\), which means either \(a_0 = 0\) or \(r^2 + r + 1 = 0\). The radical of the indicial equation is given by: \(r_1 = 0 \text{ and } r_2 = 0\)

Find One Solution for \(x > 0\)

Using the roots \(r_1\) and \(r_2\), we can now find a series solution for \(x > 0\). Given the roots are equal, we can create the following solution: $$ J_0(x) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n} (n!)^2} $$

Show Series Convergence

In order to show that the series converges for all values of \(x\), examine the ratio of consecutive terms of the series: $$ \frac{a_{n+1}}{a_n} = \frac{(-1)^{n+1} x^{2(n+1)}}{2^{2(n+1)}((n+1)!)^2} \cdot \frac{2^{2n}(n!)^2}{(-1)^n x^{2n}} $$ Simplifying: $$ \frac{a_{n+1}}{a_n} = \frac{-x^2}{(2n+2)(2n+1)} $$ Notice that the limit as \(n\) tends towards infinity is: $$ \lim_{n \rightarrow \infty} \left|\frac{-x^2}{(2n+2)(2n+1)} \right| \leq 1 $$ Since this limit is always less than or equal to 1, the series converges for all \(x\) using the Ratio Test. Therefore, the series solution converges for all x, and the Bessel function of the first kind of order zero, \(J_0(x)\), is defined as: $$ J_{0}(x)=1+\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2 n}}{2^{2 n}(n !)^{2}} $$

Key concepts

Regular Singular Point

In the Bessel equation, a regular singular point is a unique feature where certain conditions are met. The point we are examining here is \(x = 0\). To establish this, we need to ensure:

• The coefficients of the differential equation are analytic at \(x = 0\).
• The point must not be an ordinary point.

For the equation \(x^2 y'' + x y' + x^2 y = 0\), all coefficients \(x^2\), \(x\), and \(x^2\) are indeed analytic at \(x = 0\). This means they can be written as an infinite series with a positive radius of convergence. Additionally, at \(x=0\), the coefficients of \(y'\) and \(y''\) are not non-zero, making it a regular, not ordinary, point. Hence, \(x = 0\) is confirmed as a regular singular point of the Bessel equation.

Indicial Equation

The indicial equation helps us determine the behavior of solutions near a regular singular point. To derive it, we assume a series solution \(y(x) = \sum_{n=0}^{\infty} a_n x^{n+r}\). The derivatives then incorporate these terms and shift the function by order of the series.Substituting these expressions back into the Bessel equation leads us to: \((n + r)(n + r - 1)a_n + (n + r)a_n + a_n = 0\).Focusing on the implications for \(n = 0\), we simplify this to \[(r^2 + r + 1)a_0 = 0.\]This equation suggests either \(a_0 = 0\) or the roots \(r\) satisfy \(r^2 + r + 1 = 0\). Therefore, for this case, we find that the roots are \(r_1 = r_2 = 0\). These roots indicate the starting points for constructing solutions around the singular point.

Series Solution

Once we have the roots of the indicial equation, constructing a series solution involves expanding around these roots. With both roots being zero (\(r_1 = r_2 = 0\)), we look for a solution in the form of a series like this:\[ J_0(x) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n} (n!)^2} \]This particular form is derived for the positive \(x\) domain, reflecting behaviors typical in many physical and engineering contexts where Bessel functions are applied. The series is built from the repetitive cycle through each order of expansion, retaining only terms that comply with our defined recursive relationship. This series represents one of the many possible solutions, specifically tailored to match boundary conditions at infinity or specific applications such as oscillations, wave propagation, and heat conduction.

Bessel Function of the First Kind

A Bessel function of the first kind, denoted as \(J_0(x)\), represents one of the canonical solutions to the Bessel differential equation of order zero. These functions are crucial in scenarios involving cylindrical symmetry. The series solution for the Bessel function of order zero is defined as:\[ J_0(x) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n} (n!)^2} \]Bessel functions are vital in practical applications, ranging from electromagnetic waves to mechanical vibrations. The function \(J_0(x)\) specifically caters to conditions where an axis is central, such as in heat distribution in a circular object or the vibrations of a circular drum.Aside from this qualitative significance, these functions also exhibit beautiful convergence properties. By using tests like the Ratio Test, we confirm that the series form converges for all values of \(x\), making \(J_0(x)\) well-defined for all real or complex \(x\). This is why Bessel functions frequently appear in analytical solutions and numerical computations across various scientific fields.