Bessel Functions of the First Kind
Explore how Bessel order changes radial oscillation and the regular origin value.
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Bessel Functions of the First Kind
Bessel functions of the first kind, denoted as Jn(x), are solutions to Bessel's differential equation:x2y'' + xy' + (x2 - n2)y = 0. They frequently appear in physics and engineering problems, particularly those involving wave propagation and static potentials in cylindrical or spherical coordinate systems. The function Jn(x) for an integer order n can be expressed as a Taylor series around x = 0.
Worked Example
Let's calculate the expected result of J0(2) using the series expansion:
Jn(x) = ∑ k=0∞ (-1)k (x/2)2k+n / (k! (k+n)!)
For n = 0 and x = 2, the expansion simplifies because (x/2) = 1. Thus, the terms are (-1)k / (k!)2.
- Term k=0: 1 / (0!)2 = 1
- Term k=1: -1 / (1!)2 = -1
- Term k=2: 1 / (2!)2 = 0.25
- Term k=3: -1 / (3!)2 = -1 / 36 ≈ -0.0277778
- Term k=4: 1 / (4!)2 = 1 / 576 ≈ 0.0017361
Summing these first few terms gives the calculation 1 - 1 + 0.25 - 0.0277778 + 0.0017361 = 0.2239583. Continuing to infinity yields the more precise expected result 0.223890779. The alternating nature of the series demonstrates why the stable term recurrence calculation is useful.
Experiment
In the interactive module below, you can select the order n and the probe location x. Observe how the value of the Bessel function changes and how the functions oscillate.
- Set "Order n" to 0 and "Probe x" to 0 to see that the function starts at 1.
- Set "Order n" to 1 or 2 and observe that the function starts at 0 for all other integer orders.
Try changing the order to n = 1 and note the root locations. Also, check out the full version at https://riano.app/module/bessel-functions-first-kind.
Limitations
The implementation shown uses a direct term recurrence from the power series. While this method is stable and precise for small values of x (like the range [0, 6] shown here), it becomes numerically unstable for large x due to severe cancellation between large alternating terms. For larger values of x, asymptotic expansions or more sophisticated algorithms (like Miller's algorithm) must be used.
References
The DLMF provides comprehensive definitions, power series expansions, and properties of Bessel functions, serving as the primary reference for this module's mathematical implementation.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Bessel Functions of the First Kind module.
Try Bessel Functions of the First Kind on Riano →