Bessel Expansions


This section presents Bessel's expansions for functions defined on a finite interval (0,ℓ). This expansions has similar properties as Fourier series, including Gibbs phenomenon.

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Introduction to Linear Algebra with Mathematica

Bessel functions


Bessel's equation of order \( \nu \)

\[ x^2 y'' + x\,y' + \left( x^2 - \nu^2 \right) y =0 \qquad\mbox{or in self-adjoint form}\qquad \frac{\text d}{{\text d}x} \left( x\,\frac{{\text d} y}{{\text d} x} \right) + \left( x - \frac{\nu^2}{x}\right) =0 \]
has a regular singularity at x = 0. This differential equation together with boundary condition
\[ a\,y(\ell ) + b\,y' (\ell ) =0 \]
becomes a singular Sturm--Liouville problem that has eigenfunctions expressed through Bessel functions. The substitution \( y =u x^{-1/2} \) yields the reduced form of Bessel's equation
\[ u'' + \left( 1 + \frac{1- 4\nu^2}{4x^2} \right) u=0 . \]
This equation was first solved by Daniel Bernoulli (1700-1782), the second son of Johann Bernoulli, who studied the oscillations of a chain suspended by one end, and Leonhard Euler (1707--1783), who analyzed the vibrations of a stretched membrane. However, the systematic analysis of solutions to the above equation was conducted around 1817 by the German astronomer Friedrich Wilhelm Bessel (1784--1846) during an investigation of solutions of one of Kepler’s equations of planetary motion.

The coefficients of the Frobenius solution

\[ y(x) = \sum_{k\ge 0} a_k x^{k+\sigma} \]
satisfy
\[ \sum_{k\ge 0} a_k \left[ (k+\sigma )^2 - \nu^2 \right] x^{k+\sigma} + \sum_{k\ge 2} a_{k-2} x^{k+\sigma} =0. \]
The indicial equation \( \sigma^2 = \nu^2 \) has two roots \( \sigma = \pm \nu \) to which correspond two linearly independent solutions.

The solution that is finite at x = 0 can be represented by

\[ J_{\nu} (x) = \left( \frac{x}{2} \right)^{\nu} \sum_{k\ge 0} \frac{(-1)^k x^{2k}}{4^k k! \Gamma (k+\nu +1)} , \]
where Γ is the Gamma function of Euler. The above series (which converges for all real or complex x) defines the Bessel function of the first kind. If \( \nu \) is not an integer, then the function \( J_{-\nu} (x) \) is linearly independent of \( J_{\nu} (x) . \) Therefore these two functions constitute the fundamental set of solutions to Bessel's equation. When \( \nu =n \) is an integer,
\[ J_{-n} (x) = (-1)^n \,J_{n} (x) , \]
and we need to find another linearly independent solution to Bessel's equation. This can be done using reduction of order, which leads to another linearly independent solution:
\[ J_{n} (x) \,\int \frac{{\text d} x}{x \left[ J_n (x) \right]^2} . \]
Upon multiplication by a suitable constant, this expression defines either \( Y_n (x) = \pi N_n (x), \) the Weber function, or \( N_n (x) , \) the Neumann function. Heinrich Martin Weber (1842--1913) and Carl (also Karl) Gottfried Neumann (1832--1925) were German mathematicians. The Neumann function can be defined by a single expression:
\[ N_{\nu} (x) = \frac{\cos \nu\pi \,J_{\nu} (x) - J_{-\nu} (x)}{\sin \nu\pi} , \]
which is valid for all values of \( \nu . \) If \( \nu =n , \) an integer, then l'Hopital's rule yields
\begin{align*} Y_n (x) &= \pi\,N_n (x) = 2\,J_n (x) \left( \ln \frac{x}{2} + \gamma \right) - \sum_{k=0}^{n-1} \frac{(n-k-1)!}{k!} \left( \frac{x}{2} \right)^{2k-n} \\ &\quad - \sum_{k\ge 0} \frac{(-1)^k x^{2k+n}}{2^{2k+n} k! (n+k)!} \left[ 2 \left( 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) + \frac{1}{k+1} + \frac{1}{k+2} + \cdots + \frac{1}{k+n} \right] , \end{align*}
where \( \gamma = \Gamma '(1) \approx 0.5772156 \ldots \) is the Euler constant. Bessel functions can be expressed through the single Bessel function of order zero:
\[ J_n (x) = {\bf j}^n T_n \left( {\bf j}\,\frac{\text d}{{\text d}x} \right) J_0 (x) , \qquad n=1,2,\ldots , \]
where Tn is the Chebychev polynomial of the first kind and j is the unit vector in positive vertical direction on the complex plane. The parametric Bessel differential equation
\[ x^2 y'' + x\,y' + \left( \alpha^2 x^2 - \nu^2 \right) y =0 \]
has the general solution on \( (0, \infty ) : \)
\[ y(x) = A\,J_{\nu} (\alpha x ) + B\, Y_{\nu} (\alpha x ) \qquad \mbox{or} \qquad y(x) = A\,J_{\nu} (\alpha x ) + B\, N_{\nu} (\alpha x ) , \]
with arbitrary constants A and B. When \( \alpha = {\bf j} , \) the imaginary unit (\( {\bf j}^2 =-1 \) ), we get the modified Bessel equation
\[ x^2 y'' + x\,y' - \left( x^2 + \nu^2 \right) y =0 \qquad\mbox{or in self-adjoint form}\qquad \frac{\text d}{{\text d}x} \left( x\,\frac{{\text d} y}{{\text d} x} \right) - \left( x + \frac{\nu^2}{x}\right) =0 \]
that has two linearly independent solutions:
\begin{align*} I_{\nu} (x) &= {\bf j}^{-\nu} \,J_{\nu} \left( {\bf j} x \right) = \sum_{k\ge 0} \frac{1}{k! \,\Gamma (k + \nu +1)} \left( \frac{x}{2} \right)^{2k+\nu} , \\ K_{\nu} (x) &= \frac{\pi}{2} \,\frac{I_{-\nu}(x) - I_{\nu} (x)}{\sin \pi\nu} , \end{align*}
called the modified Bessel functions of the first and second kind. The function \( K_{\nu} (x) \) is also referred to as Macdonald's function.

In 1873, Hermite used a sequence of polynomials \( \{ y_n (x) \}_{n\ge 0} \) in his proof of the transcendency of e. Also, in 1880, a German physicist Heinrich Rudolf Hertz (1857--1894) essentially introduced a Bessel polynomial of degree n:

\[ y_n (x) = \left( \frac{\pi x}{2} \right)^{-1/2} e^{1/x} \,K_{n + 1/2} \left( \frac{1}{x} \right) = \sum_{k=0}^n \frac{(n+k)!}{k! \,(n-k)!}\left( \frac{x}{2} \right)^k , \]
where \( K_{n+ 1/2} (x) \) is Macdonald's function. The Bessel polynomials satisfy the differential equation
\[ x^2 y'' + 2 \left( x+1 \right) y' -n \left( n+1 \right) y =0 . \]
From the formula
\[ \frac{\text d}{{\text d}x} \left( x^2 e^{-2/x}\left( y'_n y_m - y_m y'_m \right) \right) = \left( n-m \right) \left( n_m+1 \right) e^{-2/x} \,y_n y_m , \]
it follows that there is no ordinary function ψ that satisfy the orthogonality relation
\[ \int_{-\infty}^{\infty} y_n (x)\,y_m (x)\,{\text d} \psi (x) = K_n \delta_{n,m} , \quad k_n \ne 0, \quad n,m = 0,1,2,\ldots , \]
but the generating function (distribution):
\[ \frac{{\text d}\psi}{{\text d}x} = w(x) , \quad \mbox{where} \quad w(x) = - \sum_{n\ge 0} \frac{2^{n+1} \delta^{(n)} (x)}{n! \,(n+1)!} . \]

If we are looking for bounded solutions on the finite interval \( (0, \ell ) , \) we have to disregard the Weber/Neumann function (by setting the corresponding coefficient to zero) because these functions are unbounded at the origin. Therefore, its solution will be \( J_{\nu} (\alpha x) . \)

There are known three typical types of boundary conditions:

 

Bessel--Fourier Series


The Fourier--Bessel series of a function f(x) defined in the finite interval \( (0, \ell ) \) is its representation as infinite series

\[ f (x) = \sum_{n \ge 1} c_n J_{\nu} \left( \mu_{\nu , n} \,\frac{x}{\ell} \right) , \]
where \( \mu_{\nu , n} \) (which we will denote by \( \alpha_{n} \) dropping index ν) is a root, numbered n associated with the Bessel function \( J_{\mu} \) and cn are the assigned coefficients associated with one of three types boundary conditions. Let αn (\( n=1,2, \ldots ) \) be a sequence of roots of one of the types boundary conditions (Dirichlet, Neumann, or third kind), then \( \lambda_n = \alpha_n^2 \) becomes the eigenvalue of the corresponding Sturm--Liouville problem, with eigenfunction \( J_{\nu} \left( \alpha_n \,\frac{x}{\ell} \right) . \)
  1. Dirichlet boundary conditions: \( y(\ell ) =0 \qquad \Longrightarrow \qquad J_{\nu} (\alpha \ell ) =0. \)
    \[ c_n = \frac{2}{\ell^2 J^2_{\nu +1} (\alpha_n \ell )} \, \int_0^{\ell} x\,f(x)\, J_{\nu} (\alpha_n x)\,{\text d}x, \qquad n=1,2,\ldots . \]
  2. Neumann boundary conditions: \( y'(\ell ) =0 \qquad \Longrightarrow \qquad J_{\nu}' (\alpha \ell ) =0. \)
    \[ c_n = \frac{2}{\ell^2 J^2_{\nu} (\alpha_n \ell )} \, \int_0^{\ell} x\,f(x)\, J_{\nu} (\alpha_n x)\,{\text d}x, \qquad n=1,2,\ldots . \]
  3. Boundary conditions of third kind: \( h\,y(\ell ) + y' (\ell )=0 \qquad \Longrightarrow \qquad h\,J_{\nu} (\alpha \ell ) + \alpha \,J_{\nu}' (\alpha \ell ) =0. \)
    \[ c_n = \frac{2\alpha_n^2}{\left( \ell^2 \alpha_n^2 -\nu^2 +h \right) J^2_{\nu} (\alpha_n \ell )} \, \int_0^{\ell} x\,f(x)\, J_{\nu} (\alpha_n x)\,{\text d}x, \qquad n=1,2,\ldots . \]
If the Fourier series of f converges at the point x, then the Fourier--Bessel series of f converges to the same value at the same point.

Example: Consider the function \( f(x) = 1- x^3 \) on the interval [0,1]. Suppose we want to approximate this functions by Bessel polynomial associated with the roots from the Dirichlet boundary conditions.

First, we find the roots of the Bessel function of order 0:

zeros = N[BesselJZero[0, Range[50]]]
Out[10]=
{2.40483, 5.52008, 8.65373, 11.7915, 14.9309, 18.0711, 21.2116, \
24.3525, 27.4935, 30.6346, 33.7758, 36.9171, 40.0584, 43.1998, \
46.3412, 49.4826, 52.6241, 55.7655, 58.907, 62.0485, 65.19, 68.3315, \
71.473, 74.6145, 77.756, 80.8976, 84.0391, 87.1806, 90.3222, 93.4637, \
96.6053, 99.7468, 102.888, 106.03, 109.171, 112.313, 115.455, \
118.596, 121.738, 124.879, 128.021, 131.162, 134.304, 137.446, \
140.587, 143.729, 146.87, 150.012, 153.153, 156.295}

Table of square norms of the Bessel functions:

Table[(BesselJ[1, zeros[[k]]])^2 /2, {k, 1, 10}]
Out[2]= {0.134757, 0.0578901, 0.0368432, 0.0270188, 0.0213307, 0.0176211, \ 0.0150105, 0.0130737, 0.0115796, 0.0103919}
For the function \( f(x) = 1-x^3, \) we calculates its Fourier--Bessel approximation with N=5 terms:
sum0[x_] =
Sum[2/(BesselJ[1, zeros[[k]]])^2*BesselJ[0, x*zeros[[k]]]*
NIntegrate[x*BesselJ[0, x*zeros[[k]]]*(1 - x^3), {x, 0, 1}], {k, 1,
5}]
Out[3]=
1.27215 BesselJ[0, 2.40483 x] - 0.334787 BesselJ[0, 5.52008 x] +
0.0959387 BesselJ[0, 8.65373 x] - 0.0483534 BesselJ[0, 11.7915 x] +
0.025496 BesselJ[0, 14.9309 x]

Then we calculate its mean square error:

NIntegrate[((1 - x^3) - sum0[x])^2 , {x, 0, 1}]
Out[4]= 0.000018408

and plot

Plot[sum0[x], {x, 0, 1}, PlotStyle -> {Thick, Black}]

Next we repeat the same Fourier--Bessel approximation using the Bessel function of order 1.

zeros1 = N[BesselJZero[1, Range[50]]]
Out[6]=
{3.83171, 7.01559, 10.1735, 13.3237, 16.4706, 19.6159, 22.7601, \
25.9037, 29.0468, 32.1897, 35.3323, 38.4748, 41.6171, 44.7593, \
47.9015, 51.0435, 54.1856, 57.3275, 60.4695, 63.6114, 66.7532, \
69.8951, 73.0369, 76.1787, 79.3205, 82.4623, 85.604, 88.7458, \
91.8875, 95.0292, 98.171, 101.313, 104.454, 107.596, 110.738, \
113.879, 117.021, 120.163, 123.304, 126.446, 129.588, 132.729, \
135.871, 139.013, 142.154, 145.296, 148.438, 151.579, 154.721, \
157.863}
Table[(BesselJ[2, zeros1[[k]]])^2 /2, {k, 1, 10}]
Out[7]=
{0.0811076, 0.0450347, 0.0311763, 0.0238404, 0.0192993, 0.0162114, \
0.0139753, 0.0122814, 0.0109536, 0.009885}
sum1[x_] =
Sum[2/(BesselJ[2, zeros1[[k]]])^2*BesselJ[1, x*zeros1[[k]]]*
NIntegrate[x*BesselJ[1, x*zeros1[[k]]]*(1 - x^3), {x, 0, 1}], {k,
1, 5}]
Out[8]=
1.6444 BesselJ[1, 3.83171 x] + 0.00715804 BesselJ[1, 7.01559 x] +
0.717818 BesselJ[1, 10.1735 x] - 0.104772 BesselJ[1, 13.3237 x] +
0.503362 BesselJ[1, 16.4706 x]
NIntegrate[((1 - x^3) - sum1[x])^2 , {x, 0, 1}]
Out[9]=
0.0493142
Plot[sum1[x], {x, 0, 1}, PlotStyle -> {Thick, Black}]
This approximation cannot be good in the neighborhood of the origin because the given function is 1 at end point x = 0 but the Bessel function of order 1 is zero: \( J_1 (0) =0. \)

Example: Let us consider the function \( f(x) = x(3-x)^2 \) on the interval [0,3]. This function satisfies the homogeneous Neumann condition at right end point x =3 and the homogeneous Dirichlet condition at the origin. We expand the function into two Bessel series with respect to Bessel function of order zero and 2:

\begin{align*} x(3-x)^2 &= a_0 + \sum_{n\ge 1} a_n \,J_0 \left( \alpha_n \,\frac{x}{3} \right) , \\ x(3-x)^2 &= b_0 + \sum_{n\ge 1} b_n \,J_2 \left( \beta_n \,\frac{x}{3} \right) , \end{align*}
where αn are roots of the transcendent equation \( J'_0 (\alpha ) =0 \) and βn are roots of the equation \( J'_2 (\beta ) =0 . \) Hence,
\begin{align*} a_n &= \frac{2}{3^2 J_0^2 (\alpha_n )} \, \int_0^2 x(2-x)^2 \,J_0 \left( \alpha_n \,\frac{x}{3} \right) {\text d} x, \qquad n=1,2,\ldots , \\ b_n &= \frac{2}{3^2 J_2^2 (\beta_n )} \, \int_0^2 x(2-x)^2 \,J_2 \left( \beta_n \,\frac{x}{3} \right) {\text d} x, \qquad n=1,2,\ldots . \end{align*}

Example: Consider the shifted Heaviside function \( f(t) = H(t-1) \) on the interval [0,2]. We expand this function into Fourier--Bessel series with respect to the roots of the equation of the third kind: \( J_0 (2\alpha ) + \alpha \,J'_0 (\alpha ) =0: \)

\[ H(t-1) = \sum_{n\ge 1} c_n \, J_{0} (\alpha_n x) , \]
where
\[ c_n = \frac{2\alpha_n^2}{\left( 2^2 \alpha_n^2 +1 \right) J^2_{0} (\alpha_n 2 )} \, \int_1^{2} x\, J_{0} (\alpha_n x)\,{\text d}x, \qquad n=1,2,\ldots . \]

Example: Consider the shifted Dirac delta function (which is actually is a distribution)

\[ \delta (t-a) = x\,\int_{0}^{\infty} J_{\nu} (xt) \, J_{\nu} (at)\, t{\text d}t , \qquad \Re \nu > -1, \quad x > 0, \quad a> 0. \]
The Dirac function can represented via the Airy functions:
\[ \delta (t-a) = \int_{-\infty}^{\infty} \mbox{Ai} (t-x) \, \mbox{Ai} (t-a)\, {\text d}t . \]

 

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