Question

Consider the Bessel equation of order \(v\) $$ x^{2} y^{\prime \prime}+x y^{\prime}+\left(x^{2}-v^{2}\right)=0, \quad x>0 $$ Take \(v\) real and greater than zero. (a) Show that \(x=0\) is a regular singular point, and that the roots of the indicial equation are \(v\) and \(-v\). (b) Corresponding to the larger root \(v\), show that one solution is $$ y_{1}(x)=x^{v}\left[1+\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !(1+v)(2+v) \cdots(m-1+v)(m+v)}\left(\frac{x}{2}\right)^{2 m}\right] $$ (c) If \(2 v\) is not an integer, show that a second solution is $$ y_{2}(x)=x^{-v}\left[1+\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !(1-v)(2-v) \cdots(m-1-v)(m-v)}\left(\frac{x}{2}\right)^{2 m}\right] $$ Note that \(y_{1}(x) \rightarrow 0\) as \(x \rightarrow 0,\) and that \(y_{2}(x)\) is unbounded as \(x \rightarrow 0\). (d) Verify by direct methods that the power series in the expressions for \(y_{1}(x)\) and \(y_{2}(x)\) converge absolutely for all \(x\). Also verify that \(y_{2}\) is a solution provided only that \(v\) is not an integer.

Short answer

In summary: (a) \(x=0\) is a regular singular point, and the roots of the indicial equation are \(r_1 = v\) and \(r_2 = -v\). (b) The solution corresponding to the larger root \(v\) is: $$ y_1(x) = x^v \left[1 + \sum_{m=1}^{\infty} \frac{(-1)^{m}}{m!(1+v)(2+v) \cdots (m-1+v)(m+v)}\left(\frac{x}{2}\right)^{2m}\right] $$ (c) The second solution, under the condition that \(2v\) is not an integer, is: $$ y_2(x) = x^{-v} \left[1+\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !(1-v)(2-v) \cdots (m-1-v)(m-v)}\left(\frac{x}{2}\right)^{2m}\right] $$ (d) Both power series in the expressions for \(y_1(x)\) and \(y_2(x)\) have absolute convergence, and \(y_2(x)\) is a valid solution only when \(v\) is not an integer.

Step-by-step solution

(a) Identifying Regular Singular Point and Roots of Indicial Equation

To determine if \(x=0\) is a regular singular point, we need to check if the coefficients of the Bessel equation are analytic at \(x=0\). We can rewrite the given equation as: $$ y''+\frac{1}{x}y'+\frac{x^2-v^2}{x^2}y=0 $$ At \(x=0\), the second and third coefficients have well-defined Taylor series, while the first coefficient (\(1/x\)) has a simple pole. Hence, \(x=0\) is a regular singular point. Now, we need to find the roots of the indicial equation. The indicial equation for the above form is: $$ r(r-1)+r=v^2 $$ Simplifying the equation, we get: $$ r^2 = v^2 $$ Hence, the roots of the indicial equation are \(r_1 = v\) and \(r_2 = -v\).

(b) Finding Solution Corresponding to the Larger Root \(v\)

We will use Frobenius Method to construct the solution corresponding to the root \(r_1 = v\). We assume the solution is of the form: $$ y_1(x) = x^v \sum_{n=0}^{\infty} a_n x^n $$ Substituting this expression into the Bessel equation and differentiating, we get the equation: $$ \sum_{n=0}^{\infty} a_n [n(n+v-1)+n+v^2]x^{n+v-2}=0 $$ Setting the coefficients of each power of \(x\) to zero, we obtain the recurrence relation: $$ a_{n+1} = \frac{-a_n}{(n+1)(n+2v)} $$ Therefore, the solution corresponding to the root \(v\) is: $$ y_1(x) = x^v \left[1 + \sum_{m=1}^{\infty} \frac{(-1)^{m}}{m!(1+v)(2+v) \cdots (m-1+v)(m+v)}\left(\frac{x}{2}\right)^{2m}\right] $$

(c) Finding the Second Solution \(y_2(x)\)

For the second solution, we will use the Frobenius Method again to construct the solution corresponding to the root \(r_2 = -v\). Under the condition that \(2v\) is not an integer, we can assume the solution is of the form: $$ y_2(x) = x^{-v} \sum_{n=0}^{\infty} b_n x^n $$ Following the same process as in part (b), we obtain the recurrence relation: $$ b_{n+1} = \frac{-b_n}{(n+1)(n-2v)} $$ Thus, the second solution \(y_2(x)\) is: $$ y_2(x) = x^{-v} \left[1+\sum_{m=1}^{\infty} \frac{(-1)^{m}}{m !(1-v)(2-v) \cdots (m-1-v)(m-v)}\left(\frac{x}{2}\right)^{2m}\right] $$

(d) Verifying Absolute Convergence and Validity of \(y_2(x)\) as a Solution

To verify the absolute convergence of the power series in the expressions for \(y_1(x)\) and \(y_2(x)\), we will use the Ratio Test. For both series, the ratio test yields: $$ \lim_{m \to \infty} \frac{a_{m+1}}{a_m} = \lim_{m \to \infty} \frac{b_{m+1}}{b_m} = \frac{x^2}{(m+1)(m+2|v|)} $$ Since \(\lim_{m \to \infty} \frac{x^2}{(m+1)(m+2|v|)} < 1\) for all \(x\), both series converge absolutely. Now, we will check if \(y_2(x)\) is a solution only when \(v\) is not an integer. If \(v\) were an integer, then one of the denominators in the series for \(y_2(x)\) would be zero, making it undefined. As the condition \(2v\) is not an integer essentially means that \(v\) is not an integer, it implies that \(y_2(x)\) is a valid solution only under this condition.

Key concepts

Regular Singular Point

The Bessel equation showcases a type of differential equation that often arises in the sciences, especially in physics, with applications stretching from the description of waves to the quantum mechanics realm. In trying to solve it, the concept of a regular singular point comes to the fore.

A point is designated as this type of singular point if certain conditions regarding the behavior of the coefficients of the equation are met when they are approached. In simpler terms, at a regular singular point, some terms in the equation may go to infinity, but not so wildly that finding a solution around that point becomes impossible. For the given Bessel equation, the point at x=0 is of interest. Through meticulous analysis, one can ascertain that as x approaches zero, the coefficients of the equation do not behave 'nicely', but they do so in a 'controlled' manner, thus identifying x=0 as a regular singular point. This peculiarity allows use of specialized techniques like the Frobenius Method to find a power series solution around the singular point.

Indicial Equation

Upon identifying a regular singular point, the next logical step in unraveling the Bessel equation is to work with the so-called indicial equation. This is a fundamental step in the Frobenius Method, which is a strategy to look for solutions near a regular singular point.

The indicial equation essentially represents the leading behavior of the power series solution near the singular point. It is derived by focusing on the lowest powers of x within the differential equation because these dictate the behavior near the singularity. The indicial equation is obtained by plugging in a trial solution into the original equation and isolating the terms with the lowest powers of x. The roots of this indicial equation, which for the Bessel equation are v and -v, provide initial conditions for the power series, setting the stage for further calculations with the Frobenius Method toward a full solution.

Frobenius Method

Diving deeper into the Frobenius Method, we find a technique designed to deal with differential equations having a regular singular point - a lifeline for equations like Bessel's. This method involves expressing the solution as a power series with coefficients that need to be determined. However, the power series begins with the exponent corresponding to the roots of the indicial equation instead of the usual zero or one.

The crux of the Frobenius Method lies in the systematic exploitation of the recurrence relations that arise when substituting this power series back into the differential equation. By equating coefficients of like powers of x, a series of relations between the coefficients of the power series can be discerned, enabling us to recursively calculate them. Indeed, for Bessel's equation, these recurrence relations are pivotal to determining the coefficients of the series that form part of the sought-after solutions.

Power Series Solution

The power series solution is where the Frobenius Method yields its fruits. Through careful arithmetic and attention to detail, we can construct a solution to our Bessel equation in the form of an infinite series. This power series solution not only testifies to the elegance of the Frobenius Method but also speaks volumes about the method's utility in addressing differential equations with singular points.

In the case of the Bessel equation, two possible power series solutions emerge after laboriously applying the Frobenius Method. The first series solution corresponds to the larger root v, and the other to the negative root -v. The first series winds down to zero as x approaches the singular point, indicating a noteworthy behavior of the Bessel function near the origin. On the other hand, the second solution becomes unbounded, painting a diverse picture of the function's nature and ensuring that the full spectrum of solutions is encompassed.

Absolute Convergence

Finally, when entertaining infinite series, the question of convergence is crucial. Specifically, we aim for a type of convergence that assures us the series genuinely represents a function and does so in a well-behaved manner—this is termed absolute convergence.

Absolute convergence means that the sum of the absolute values of the series' terms converges, guaranteeing that the series does not merely converge due to the cancellation of large, opposite terms. For the power series uncovered during our Bessel exploration, we call upon the Ratio Test, a reliable sentinel guarding the gates to convergence. The Ratio Test checks the ratios of successive terms, and if the limit of these ratios as the terms approach infinity is less than one, the series converges absolutely. The power series emerging from the Bessel equation pass this test with flying colors, strengthening our confidence in the series as genuine representatives of solutions.