Bessel function guides

This cluster of ten articles walks from what a Bessel function is through formulas, calculation methods, worked examples, J_v and Y_v, modified functions, graphs, and physics applications. Each post links to the Bessel Function Calculator on the home page so you can check numbers and plots as you read.

All articles

Bessel Functions in Physics: Applications
Bessel functions in waveguides, vibrating disks, quantum mechanics on a disk, circular diffraction, and steady heat flow, with typical numbers you can check online.
Bessel Function Graphs: Zeros and Orders
How to read J_v plots, locate zeros and peaks, compare orders on the same axis, and use the calculator's local graph for quick homework sanity checks.
Modified Bessel Functions I_v(x) and K_v(x)
Definitions of modified Bessel I_v and K_v, exponential growth and decay, the modified equation, and how they connect to cylindrical J_v through ix.
Bessel Function of the Second Kind Y_v(x)
Understand Y_v(x), its singularity at the origin, annulus boundary problems, and when Y_v must appear beside J_v in cylindrical wave and field solutions.
Bessel Function of the First Kind J_v(x)
What J_v(x) means for radial modes, how order v shapes the curve, essential identities, and how to check J_v values with the free Bessel function calculator.
Bessel Function Examples: Worked Problems
Worked J_n(x) values, disk and waveguide mode problems, recurrence checks, and zero conditions you can verify in under a minute with the online calculator.
How to Use the Bessel Function Calculator
Step-by-step guide to real and complex mode, entering order v and argument x, reading J_v tables, and using the local plot on the Bessel Function Calculator home page.
How to Calculate Bessel Functions: Methods Guide
Compare series, recurrence, asymptotics, and library methods for J_v(x). Learn which approach fits small x, large x, and complex arguments, then spot-check online.
Bessel Function Formula: Equation and Identities
Bessel's differential equation, power series for J_n(x), recurrence relations, and derivative formulas. Verify each identity with the free Bessel function calculator.
What Is a Bessel Function? Definition and Uses
Learn what Bessel functions are, why they appear in circular and cylindrical problems, and how J_v(x) behaves at the origin, at zeros, and in real engineering models.