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Introduction to Linear Algebra with Mathematica

Glossary

Preface

In this section, we show that Bessel's functions

\[ \phi_n (x) = J_{\nu} \left( \mu_n \frac{x}{\ell} \right) \qquad (n=1,2,3,\ldots ) \]
are orthogonal when parameters μn are positive roots of some transcendent equation involving Bessel functions of the first kind. Orthogonal means that
\[ \left\langle \phi_n (x) , \phi_k (x) \right\rangle = \int_0^{\ell} J_{\nu} \left( \mu_n \frac{x}{\ell} \right) J_{\nu} \left( \mu_k \frac{x}{\ell} \right) x\,{\text d}x = \begin{cases} 0, & \ \mbox{ if } \quad n \ne k , \\ \| J_{\nu} \|^2 , & \ \mbox{ when } \quad n=k , \end{cases} \]
where the value of the norm squared, \( \| J_{\nu} \|^2 , \) depends on the boundary condition at the right endpoint x = ℓ.

Orthogonality of Bessel's functions

For any real number α ∈ ℝ, the Bessel equation with a parameter
\begin{equation} \label{EqOrtho.1} x^2 y'' + x\,y' + \left( \alpha^2 x^2 - \nu^2 \right) y = 0 \qquad \mbox{or in self-adjoint form} \qquad \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d}y}{{\text d}x} \right) + \left( \alpha^2 x - \frac{\nu^2}{x} \right) y = 0 \end{equation}
has a bounded solution
\[ \phi (x) = J_{\nu} \left( \alpha \,x \right) , \]
which can be justified by direct substitution. For two distinct positive numbers k1 and k2, we consider two functions
\[ \phi_1 (x) = J_{\nu} \left( k_1 \,x \right) \qquad \mbox{and} \qquad \phi_2 (x) = J_{\nu} \left( k_2 \,x \right) . \]
They are solutions of equations
\[ \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_1 (x)}{{\text d}x} \right) + \left( k_1^2 x - \frac{\nu}{x} \right) \phi_1 (x) = 0 \]
and
\[ \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_2 (x)}{{\text d}x} \right) + \left( k_2^2 x - \frac{\nu}{x} \right) \phi_2 (x) = 0 \]
respectively. Multiplying the forme by ϕ2(x) and the latter by ϕ1(x), and subtracting the results, we obtain
\[ \left( k_1^2 - k_2^2 \right) \phi_1 (x)\,\phi_2 (x)\,x = - \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_1 (x)}{{\text d}x} \right) \phi_2 (x) + \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_2 (x)}{{\text d}x} \right) \phi_1 (x) . \]
Integrating both sides of the latter with respect to x ∈ [0, ℓ], we get
\[ \left( k_1^2 - k_2^2 \right) \int_0^{\ell} \phi_1 (x)\,\phi_2 (x)\,x \,{\text d}x = - \int_0^{\ell} {\text d}x\, \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_1 (x)}{{\text d}x} \right) \phi_2 (x) + \int_0^{\ell} {\text d}x\, \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_2 (x)}{{\text d}x} \right) \phi_1 (x) . \]
Performing integration by parts shows
\[ - \int_0^{\ell} {\text d}x\, \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_1 (x)}{{\text d}x} \right) \phi_2 (x) + \int_0^{\ell} {\text d}x\, \frac{\text d}{{\text d}x} \left( x\, \frac{{\text d} \phi_2 (x)}{{\text d}x} \right) \phi_1 (x) = \left. x\, \frac{{\text d} \phi_2 (x)}{{\text d}x} \, \phi_1 (x) - x\, \frac{{\text d} \phi_1 (x)}{{\text d}x} \, \phi_2 (x) \right\vert_{x=0}^{x=\ell} . \]
If ν > 1, the lower limit becomes zero, and we get
\[ \left( k_1^2 - k_2^2 \right) \int_0^{\ell} \phi_1 (x)\,\phi_2 (x)\,x \,{\text d}x = \ell\left. \frac{{\text d} \phi_2 (x)}{{\text d}x} \right\vert_{x=\ell} \, \phi_1 (\ell ) - \ell\, \left. \frac{{\text d} \phi_1 (x)}{{\text d}x} \right\vert_{x=\ell} \, \phi_2 (\ell ) \]
Upon setting k1 = μn/ℓ and k2 = μk/ℓ, we obtain the integral relation
\begin{equation} \label{EqOrtho.2} \frac{\left( \mu_n^2 - \mu_k^2 \right)}{\ell^2} \int_0^{\ell} {\text d}x \,x\,J_{\nu} \left( \mu_n \frac{x}{\ell}\right) J_{\nu} \left( \mu_k \frac{x}{\ell}\right) = \mu_k J_{\nu} \left( \mu_n \right) J'_{\nu} \left( \mu_k \right) - \mu_n J_{\nu} \left( \mu_k \right) J'_{\nu} \left( \mu_n \right) . \end{equation}
If parameters μn and μk are chosen in a way to annihilate the right-hand side of Eq.\eqref{EqOrtho.2}, we get orthogonality of Bessel's functions. We consider three important cases of boundary conditions for which Bessel's functions are orthogonal.

 

Dirichlet boundary conditions

Let μn (n = 1, 2, 3, …) be a sequence of positive roots of the equation
\[ J_{\nu} (\mu ) = 0. \]
Then right-hand side of Eq.\eqref{EqOrtho.2} wil be zero for n ≠ k. So we need to determine
\[ \left\| J_{\nu} \left( \mu_n \frac{x}{\ell}\right) \right\|^2 = \left\langle J_{\nu} \left( \mu_n \frac{x}{\ell}\right) , J_{\nu} \left( \mu_n \frac{x}{\ell}\right) \right\rangle = \int_0^{\ell} J_{\nu}^2 \left( \mu_n \frac{x}{\ell}\right) x\,{\text d}x . \]
We find its value by taking the limit as k → μn in the orthogonality relation \eqref{EqOrtho.2}:
\[ \| J_{\nu} \|^2 = \lim_{k\to \mu_n} \,\frac{\ell^2}{k^2 - \mu_n^2} \left[ \mu_n J_{\nu} \left( k \right) J'_{\nu} \left( \mu_n \right) - k\, J_{\nu} \left( \mu_n \right) J'_{\nu} \left( k \right) \right] \]
Application of the l'Hôpital's rule yields
\[ \| J_{\nu} \|^2 = \lim_{k\to \mu_n} \frac{\ell^2}{2k} \,\frac{\text d}{{\text d}k} \left\{ \mu_n J_{\nu} \left( k \right) J'_{\nu} \left( \mu_n \right) \right\} = \frac{\ell^2}{2}\, \left[ J'_{\nu} \left( \mu_n \right) \right]^2 = \frac{\ell^2}{2}\, \left[ J_{\nu +1} \left( \mu_n \right) \right]^2 . \]
Thus, we have
\begin{equation} \label{EqOrtho.3} \left\langle J_{\nu} \left( \mu_n \frac{x}{\ell}\right) , J_{\nu} \left( \mu_k \frac{x}{\ell}\right) \right\rangle = \begin{cases} 0, & \ \mbox{ if } \quad n\ne k , \\ \frac{\ell^2}{2}\,J_{\nu +1} \left( \mu_n \right) , & \ \mbox{ when }\quad n=k. \end{cases} \end{equation}

 

Neumann boundary conditions

Let μn (n = 1, 2, 3, …) be a sequence of positive roots of the equation
\[ J'_{\nu} (\mu ) = 0. \]
Then the right-hand side of Eq.\eqref{EqOrtho.2} will be zero for n ≠ k. To determine the value of square norm when n = k, we again apply the l'Hôpital's rule and obtain
\begin{equation} \label{EqOrtho.4} \left\langle J_{\nu} \left( \mu_n \frac{x}{\ell}\right) , J_{\nu} \left( \mu_k \frac{x}{\ell}\right) \right\rangle = \begin{cases} 0, & \ \mbox{ if } \quad n\ne k , \\ \frac{\ell^2}{2}\,J_{\nu} \left( \mu_n \right) , & \ \mbox{ when }\quad n=k. \end{cases} \end{equation}

 

Boundary conditions of the third kind

Let μn (n = 1, 2, 3, …) be a sequence of positive roots of the equation
\[ a \ell\,J_{\nu} (\mu ) + b\,\mu\,J'_{\nu} (\mu ) =0 , \]
where 𝑎 and b are some real numbers. It is not hard to verify that the right-hand side of Eq.\eqref{EqOrtho.2} will be zero for n ≠ k. To determine the value of square norm when n = k, we take the limit
\[ \| J_{\nu} \|^2 = \int_0^{\ell} J_{\nu}^2 \left( \mu_n \frac{x}{\ell} \right) x\,{\text d}x = \lim_{k\to \mu_n} \,\frac{\ell^2}{k^2 - \mu_n^2} \left[ \mu_n J_{\nu} \left( k \right) J'_{\nu} \left( \mu_n \right) - k\, J_{\nu} \left( \mu_n \right) J'_{\nu} \left( k \right) \right] \]
We again apply the l'Hôpital's rule and obtain
\begin{align*} \| J_{\nu} \|^2 &= \frac{\ell^2}{2\,\mu_n}\lim_{k\to \mu_n} \frac{\text d}{{\text d}k} \left\{ \mu_n J_{\nu} \left( k \right) J'_{\nu} \left( \mu_n \right) - k\, J_{\nu} \left( \mu_n \right) J'_{\nu} \left( k \right) \right\} \\ &= \frac{\ell^2}{2\,\mu_n} \lim_{k\to \mu_n} \left\{ \mu_n J'_{\nu} \left( k \right) J'_{\nu} \left( \mu_n \right) - J_{\nu} \left( \mu_n \right) J'_{\nu} \left( k \right) - k\, J_{\nu} \left( \mu_n \right) J''_{\nu} \left( k \right) \right\} \\ &= \frac{\ell^2}{2\,\mu_n} \left\{ \mu_n J'_{\nu} \left( \mu_n \right) J'_{\nu} \left( \mu_n \right) - J_{\nu} \left( \mu_n \right) J'_{\nu} \left( \mu_n \right) - \mu_n J_{\nu} \left( \mu_n \right) J''_{\nu} \left( \mu_n \right) \right\} . \end{align*}
From Bessel's equation, we have
\[ -\mu\, J''_{\nu} (\mu ) = J'_{\nu} (\mu ) + \left( \mu - \frac{\nu^2}{\mu} \right) J_{\nu} (\mu ) . \]
So
\[ -\mu\,J_{\nu} (\mu )\,J''_{\nu} (\mu ) = J_{\nu} (\mu )\,J'_{\nu} (\mu ) + \left( \mu - \frac{\nu^2}{\mu} \right) J_{\nu} (\mu )\,J_{\nu} (\mu ) , \]
and we get
\[ \| J_{\nu} \|^2 = \frac{\ell^2}{2} \left\{ \left[J'_{\nu} (\mu_n ) \right]^2 + \left( 1 - \frac{\nu^2}{\mu^2_n} \right) J_{\nu}^2 (\mu_n ) \right\} \]

 

  1. Bowman, Frank Introduction to Bessel Functions (Dover: New York, 1958). ISBN 0-486-60462-4. QA408.B68
  2. Dutka, J., On the early history of Bessel functions, Archive for History of Exact Sciences, volume 49, pages 105–134 (1995). https://doi.org/10.1007/BF00376544
  6. Watson, G.N., A Treatise on the Theory of Bessel Functions,

 

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