Matlab custom spherical bessel function NaN at 0

Resolving NaN Issues in MATLAB's Spherical Bessel Functions at Zero

Understand why MATLAB's besselj and bessely functions might return NaN for spherical Bessel functions at x=0 and learn robust methods to handle these edge cases.

Spherical Bessel functions are crucial in many areas of physics and engineering, particularly in problems involving spherical coordinates, such as wave propagation, quantum mechanics, and electromagnetism. MATLAB provides functions like besselj and bessely to compute these. However, users often encounter NaN (Not a Number) results when evaluating spherical Bessel functions of the second kind (often denoted as y_n(x) or n_n(x)) at x=0. This article delves into the mathematical reasons behind this behavior and provides practical solutions for handling these edge cases in MATLAB.

Understanding Spherical Bessel Functions and Their Behavior at Zero

Spherical Bessel functions are directly related to ordinary Bessel functions. The spherical Bessel function of the first kind, j_n(x), and the second kind, y_n(x) (also known as spherical Neumann function), are defined as:

j_n(x) = \sqrt{\frac{\pi}{2x}} J_{n + 1/2}(x)

y_n(x) = \sqrt{\frac{\pi}{2x}} Y_{n + 1/2}(x)

where J_v(x) and Y_v(x) are the ordinary Bessel functions of the first and second kind, respectively, of order v.

The behavior at x=0 is critical:

• j_n(x) (Spherical Bessel of the first kind): For n=0, j_0(x) = \frac{\sin(x)}{x}. As x \to 0, j_0(x) \to 1. For n > 0, j_n(x) \to 0 as x \to 0.
• y_n(x) (Spherical Bessel of the second kind): For all n \ge 0, y_n(x) has a singularity at x=0. Specifically, y_0(x) = -\frac{\cos(x)}{x}, which approaches -\infty as x \to 0. For higher orders, y_n(x) also diverges at x=0.

MATLAB's besselj(n, x) and bessely(n, x) functions are designed for ordinary Bessel functions. To compute spherical Bessel functions, you typically use the half-integer order. When x=0 is passed to bessely(n+0.5, 0), it correctly returns NaN because the function is undefined at that point.

flowchart TD
    A[Input: x, n] --> B{Is x == 0?}
    B -- Yes --> C{Is n >= 0?}
    C -- Yes --> D{Is it spherical Bessel of 1st kind (j_n)?}
    D -- Yes --> E{Is n == 0?}
    E -- Yes --> F[Result: 1]
    E -- No --> G[Result: 0]
    D -- No --> H{Is it spherical Bessel of 2nd kind (y_n)?}
    H -- Yes --> I[Result: NaN (Singularity)]
    H -- No --> J[Error: Invalid function type]
    C -- No --> K[Error: Invalid order n]
    B -- No --> L[Compute using standard Bessel functions]

Decision flow for spherical Bessel function evaluation at x=0

Implementing Custom Spherical Bessel Functions in MATLAB

To handle the x=0 case gracefully, it's often best to implement custom functions that incorporate the known limits. This allows you to define the behavior at x=0 according to your application's requirements (e.g., using the limit, or returning a specific error/value).

function jn = sphericalbesselj(n, x)
% Custom spherical Bessel function of the first kind, j_n(x)
% Handles x=0 cases.

if x == 0
    if n == 0
        jn = 1; % Limit of j0(x) as x->0 is 1
    else
        jn = 0; % Limit of jn(x) for n>0 as x->0 is 0
    end
else
    jn = sqrt(pi./(2*x)) .* besselj(n + 0.5, x);
end

end

Custom sphericalbesselj function with x=0 handling.

function yn = sphericalbessely(n, x)
% Custom spherical Bessel function of the second kind, y_n(x)
% Returns NaN at x=0 due to singularity.

if x == 0
    yn = NaN; % y_n(x) is singular at x=0
else
    yn = sqrt(pi./(2*x)) .* bessely(n + 0.5, x);
end

end

Custom sphericalbessely function explicitly returning NaN at x=0.

Handling Vectorized Inputs and Numerical Stability

The custom functions above work well for scalar x. For vectorized inputs, a more robust approach is needed. Also, for very small x values (but not exactly zero), sqrt(pi./(2*x)) can become very large, potentially leading to numerical issues or Inf before besselj or bessely are evaluated. MATLAB's built-in besselj and bessely are generally optimized for numerical stability, but direct implementation of the sqrt(pi/(2x)) factor can be problematic for small x.

A common alternative for spherical Bessel functions is to use their direct series expansions or recurrence relations, especially for small x or specific integer n values, to avoid the x=0 singularity of the 1/sqrt(x) term.

function jn = sphericalbesselj_vec(n, x)
% Custom spherical Bessel function of the first kind, j_n(x) for vectorized x.

jn = zeros(size(x));

% Handle x == 0 cases
zero_indices = (x == 0);
if any(zero_indices)
    if n == 0
        jn(zero_indices) = 1;
    else
        jn(zero_indices) = 0;
    end
end

% Handle x ~= 0 cases
non_zero_indices = ~zero_indices;
if any(non_zero_indices)
    jn(non_zero_indices) = sqrt(pi./(2*x(non_zero_indices))) .* besselj(n + 0.5, x(non_zero_indices));
end

end

Vectorized sphericalbesselj for robust handling of x=0 within an array.